AM/FM Measurements Using Multiple Frequency of Atomic Force Microscopy

ABSTRACT

Apparatus and techniques presented combine the features and benefits of amplitude modulated (AM) atomic force microscopy (AFM), sometimes called AC mode AFM, with frequency modulated (FM) AFM. In AM-FM imaging, the topographic feedback from the first resonant drive frequency operates in AM mode while the phase feedback from second resonant drive frequency operates in FM mode. In particular the first or second frequency may be used to measure the loss tangent, a dimensionless parameter which measures the ratio of energy dissipated to energy stored in a cycle of deformation.

This application claims priority from provisional No. 61/995,905, filedApr. 23, 2014, the entire contents of which are herewith incorporated byreference.

This a continuation of application Ser. No. 14/694,980, filed Apr. 23,2015, the entire contents of which are herewith incorporated byreference.

BACKGROUND OF THE INVENTION

For the sake of convenience, the current description focuses on systemsand techniques that may be realized in a particular embodiment ofcantilever-based instruments, the atomic force microscope (AFM).Cantilever-based instruments include such instruments as AFMs, molecularforce probe instruments (1D or 3D), high-resolution profilometers(including mechanical stylus profilometers), surface modificationinstruments, chemical or biological sensing probes, and micro-actuateddevices. The systems and techniques described herein may be realized insuch other cantilever-based instruments.

An AFM is a device used to produce images of surface topography (and/orother sample characteristics) based on information obtained fromscanning (e.g., rastering) a sharp probe on the end of a cantileverrelative to the surface of the sample. Topographical and/or otherfeatures of the surface are detected by detecting changes in deflectionand/or oscillation characteristics of the cantilever (e.g., by detectingsmall changes in deflection, phase, frequency, etc., and using feedbackto return the system to a reference state). By scanning the proberelative to the sample, a “map” of the sample topography or other samplecharacteristics may be obtained.

Changes in deflection or in oscillation of the cantilever are typicallydetected by an optical lever arrangement whereby a light beam isdirected onto the cantilever in the same reference frame as the opticallever. The beam reflected from the cantilever illuminates a positionsensitive detector (PSD). As the deflection or oscillation of thecantilever changes, the position of the reflected spot on the PSDchanges, causing a change in the output from the PSD. Changes in thedeflection or oscillation of the cantilever are typically made totrigger a change in the vertical position of the cantilever baserelative to the sample (referred to herein as a change in the Zposition, where Z is generally orthogonal to the XY plane defined by thesample), in order to maintain the deflection or oscillation at aconstant pre-set value. It is this feedback that is typically used togenerate an AFM image.

AFMs can be operated in a number of different sample characterizationmodes, including contact mode where the tip of the cantilever is inconstant contact with the sample surface, and AC modes where the tipmakes no contact or only intermittent contact with the surface.

Actuators are commonly used in AFMs, for example to raster the probe orto change the position of the cantilever base relative to the samplesurface. The purpose of actuators is to provide relative movementbetween different parts of the AFM; for example, between the probe andthe sample. For different purposes and different results, it may beuseful to actuate the sample, the cantilever or the tip or somecombination of both. Sensors are also commonly used in AFMs. They areused to detect movement, position, or other attributes of variouscomponents of the AFM, including movement created by actuators.

For the purposes of the specification, unless otherwise specified, theterm “actuator” refers to a broad array of devices that convert inputsignals into physical motion, including piezo activated flexures, piezotubes, piezo stacks, blocks, bimorphs, unimorphs, linear motors,electrostrictive actuators, electrostatic motors, capacitive motors,voice coil actuators and magnetostrictive actuators, and the term“position sensor” or “sensor” refers to a device that converts aphysical parameter such as displacement, velocity or acceleration intoone or more signals such as an electrical signal, including capacitivesensors, inductive sensors (including eddy current sensors),differential transformers (such as described in co-pending applicationsUS20020175677A1 and US20040075428A1, Linear Variable DifferentialTransformers for High Precision Position Measurements, andUS20040056653A1, Linear Variable Differential Transformer with DigitalElectronics, which are hereby incorporated by reference in theirentirety), variable reluctance, optical interferometry, opticaldeflection detectors (including those referred to above as a PSD andthose described in co-pending applications US20030209060A1 andUS20040079142A1, Apparatus and Method for Isolating and MeasuringMovement in Metrology Apparatus, which are hereby incorporated byreference in their entirety), strain gages, piezo sensors,magnetostrictive and electrostrictive sensors.

In both the contact and AC sample-characterization modes, theinteraction between the probe and the sample surface induces adiscernable effect on a probe-based operational parameter, such as thecantilever deflection, the cantilever oscillation amplitude, the phaseof the cantilever oscillation relative to the drive signal driving theoscillation or the frequency of the cantilever oscillation, all of whichare detectable by a sensor. In this regard, the resultantsensor-generated signal is used as a feedback control signal for the Zactuator to maintain a designated probe-based operational parameterconstant.

In contact mode, the designated parameter may be cantilever deflection.In AC modes, the designated parameter may be oscillation amplitude,phase or frequency. The feedback signal also provides a measurement ofthe sample characteristic of interest. For example, when the designatedparameter in an AC mode is oscillation amplitude, the feedback signalmay be used to maintain the amplitude of cantilever oscillation constantto measure changes in the height of the sample surface or other samplecharacteristics.

The periodic interactions between the tip and sample in AC modes inducescantilever flexural motion at higher frequencies. Measuring the motionallows interactions between the tip and sample to be explored. A varietyof tip and sample mechanical properties including conservative anddissipative interactions may be explored. Stark, et al., have pioneeredanalyzing the flexural response of a cantilever at higher frequencies asnonlinear interactions between the tip and the sample. In theirexperiments, they explored the amplitude and phase at numerous higheroscillation frequencies and related these signals to the mechanicalproperties of the sample.

Unlike the plucked guitar strings of elementary physics classes,cantilevers normally do not have higher oscillation frequencies thatfall on harmonics of the fundamental frequency. The first three modes ofa simple diving board cantilever, for example, are at the fundamentalresonant frequency (f0), 6.19f0 and 17.5 f0. An introductory text incantilever mechanics such as Sarid has many more details. Throughcareful engineering of cantilever mass distributions, Sahin, et al.,have developed a class of cantilevers whose higher modes do fall onhigher harmonics of the fundamental resonant frequency. By doing this,they have observed that cantilevers driven at the fundamental exhibitenhanced contrast, based on their simulations on mechanical propertiesof the sample surface. This approach is has the disadvantage ofrequiring costly and difficult to manufacture special cantilevers.

The simple harmonic oscillator (SHO) model gives a convenientdescription at the limit of the steady state amplitude A of theeigenmode of a cantilever oscillating in an AC mode:

$\begin{matrix}{A = \frac{F_{0}/m}{\sqrt{( {\omega_{0}^{2} - \omega^{2}} )^{2} - ( {\omega \; {\omega_{0}/Q}} )^{2}}}} & ({SHO})\end{matrix}$

where F₀ is the drive amplitude (typically at the base of thecantilever), m is the mass, ω is the drive frequency in units ofrad/sec, ω₀ is the resonant frequency and Q is the “quality” factor, ameasure of the damping.If, as is often the case, the cantilever is driven through excitationsat its base, the SHO expression becomes

$\begin{matrix}{A = \frac{A_{drive}\omega_{0}^{2}}{\sqrt{( {\omega_{0}^{2} - \omega^{2}} )^{2} - ( {\omega_{0}\omega} )^{2}}}} & ( {{SHO}\mspace{14mu} {Amp}} )\end{matrix}$

where F₀/m has been replaced with A_(drive)ω₀ ², where A_(drive) is thedrive amplitude.

The phase angle φ is described by an associated equation

$\begin{matrix}{\varphi = {\tan^{- 1}\lbrack \frac{\omega \; \omega_{0}}{Q( {\omega_{0}^{2} - \omega^{2}} )} \rbrack}} & ( {{SHO}\mspace{14mu} {Phase}} )\end{matrix}$

When these equations are fulfilled, the amplitude and phase of thecantilever are completely determined by the user's choice of the drivefrequency and three independent parameters: A_(drive), ω₀ and Q.

In some very early work, Martin, et al., drove the cantilever at twofrequencies. The cantilever response at the lower, non-resonantfrequency was used as a feedback signal to control the surface trackingand produced a topographic image of the surface. The response at thehigher frequency was used to characterize what the authors interpretedas differences in the non-contact forces above the Si and photo-resiston a patterned sample.

Recently, Rodriguez and Garcia published a theoretical simulation of anon-contact, attractive mode technique where the cantilever was drivenat its two lowest eigen frequencies. In their simulations, they observedthat the phase of the second mode had a strong dependence on the Hamakerconstant of the material being imaged, implying that this techniquecould be used to extract chemical information about the surfaces beingimaged. Crittenden et al. have explored using higher harmonics forsimilar purposes.

There are a number of techniques where the instrument is operated in ahybrid mode where a contact mode feedback loop is maintained while someparameter is modulated. Examples include force modulation andpiezo-response imaging.

Force modulation involves maintaining a contact mode feedback loop whilealso driving the cantilever at a frequency and then measuring itsresponse. When the cantilever makes contact with the surface of thesample while being so driven, its resonant behavior changessignificantly. The resonant frequency typically increases, depending onthe details of the contact mechanics. In any event, one may learn moreabout the surface properties because the elastic response of the samplesurface is sensitive to force modulation. In particular, dissipativeinteractions may be measured by measuring the phase of the cantileverresponse with respect to the drive.

A well-known shortcoming of force modulation and other contact modetechniques is that the while the contact forces may be controlled well,other factors affecting the measurement may render it ill-defined. Inparticular, the contact area of the tip with the sample, usuallyreferred to as contact stiffness, may vary greatly depending on tip andsample properties. This in turn means that the change in resonance whilemaintaining a contact mode feedback loop, which may be called thecontact resonance, is ill-defined. It varies depending on the contactstiffness. This problem has resulted in prior art techniques avoidingoperation at or near resonance.

SUMMARY OF THE INVENTION

Cantilevers are continuous flexural members with a continuum ofvibrational modes. The present invention describes different apparatusand methods for exciting the cantilever simultaneously at two or moredifferent frequencies and the useful information revealed in the imagesand measurements resulting from such methods. Often, these frequencieswill be at or near two or more of the cantilever vibrational eigenmodes

Past work with AC mode AFMs has been concerned with higher vibrationalmodes in the cantilever, with linear interactions between the tip andthe sample. The present invention, however, is centered aroundnon-linear interactions between the tip and sample that couple energybetween two or more different cantilever vibrational modes, usually keptseparate in the case of linear interactions. Observing the response ofthe cantilever at two or more different vibrational modes has someadvantages in the case of even purely linear interactions however. Forexample, if the cantilever is interacting with a sample that has somefrequency dependent property, this may show itself as a difference inthe mechanical response of the cantilever at the different vibrationalmodes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 Preferred embodiment for probing multiple eigenmodes of acantilever.

FIG. 2 Preferred embodiment for exciting voltage-dependent motion in thecantilever probe.

FIG. 3 Preferred embodiment for probing an active device.

FIG. 4 Phase and amplitude shifts of the fundamental eigenmode with andwithout the second eigenmode being driven.

FIG. 5A-5E Images of collagen fibers taken with the preferredembodiment.

FIGS. 6 and 7 Two dimensional histogram plots of the amplitude and phasefor the first and second eigenmodes.

FIG. 8 Preferred embodiment for probing an active sample in contactwhile measuring dynamic contact properties (Dual Frequency ResonanceTracking Piezo Force Microscopy (DFRT PFM)).

FIG. 9 Resonance peaks in sweep of applied potential from dc to 2 MHz.

FIGS. 10A-10D and 11 Images of a piezoelectric sample when thecantilever potential was driven at two different frequencies, oneslightly below and the other slightly above the same contact resonancefrequency.

FIG. 12 Amplitude versus frequency and phase versus frequency curvessimultaneously measured at different frequencies.

FIG. 13 Amplitude and phase curves changing in response to varyingtip-sample interactions being driven first at two different frequenciesand then at a single frequency.

FIG. 14 Amplitude versus frequency sweeps around the second resonancemade while feeding back on the first mode amplitude.

FIG. 15-16 Amplitude versus frequency and phase versus frequency curvessimultaneous measured at different frequencies.

FIG. 17-19 Images of a piezoelectric sample when the cantileverpotential was driven at two different frequencies, one slightly belowand the other slightly above the same contact resonance frequency.

FIG. 20 Preferred embodiment of an apparatus for probing the first twoflexural resonances of a cantilever and imaging in AM mode with phase isand FM mode in accordance with the present invention.

FIG. 21 Topography of a Si-epoxy (SU8) patterned wafer imaged using theLoss Tangent technique of the present invention.

FIG. 22 Steps in calculating the corrected Loss Tangent.

FIG. 23 Direct measurement of the second mode tip-sample interactionforces.

FIG. 24 EPDH/Epoxy cryo-microtomed boundary measured at 2 Hz and 20 Hzline scan rates.

FIG. 25 Simultaneous mapping of loss tangent and stiffness of anelastomer-epoxy sandwich.

FIG. 26 Simplified measurement of the phase of the second mode for smallfrequency shifts.

FIG. 27 Effects of choosing resonant modes that are softer, matched orstiffer than the tip-sample stiffness.

FIG. 28 Extension of thermal noise measurement method to higher modes.

FIG. 29 Loss tangent measurements of a polystyrene-polypropylene spincast film as a function of the free air amplitude.

FIG. 30 Signal to noise of loss tangent estimates

FIG. 31 Comparison of phase-locked loop frequency shift and effectivefrequency shifts calculated from phase and amplitude observables.

FIG. 32 A general method for calculating mechanical properties intapping mode AFM measurements

FIG. 33 Comparison of numerical (expected) and estimated modulus andindentation measurements using the preferred method described hereversus the amplitude ratio.

FIG. 34 Optimizing the response for the generalized Hertzian model toobtain information regarding the tip shape and to make the estimationmore robust.

FIG. 35 Comparison of the modulus estimated using two preferred methodsdescribed here while varying the fundamental mode setpoint.

FIG. 36 Phasor interpretation of loss tangent and differential losstangent estimations.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 is a block diagram of a preferred embodiment of an apparatus forprobing multiple eigenmodes of a cantilever in accordance with thepresent invention. The sample 1010 is positioned below the cantileverprobe 1020. The chip of the cantilever probe 1030 is driven by amechanical actuator 1040, preferably a piezoelectric actuator, but othermethods to induce cantilever motion known to those versed in the artcould also be used. The motion of the cantilever probe 1020 relative tothe frame of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved relative to the sample 1010 by ascanning apparatus 1070, preferably a piezo/flexure combination, butother methods known to those versed in the art could also be used.

The motion imparted to the cantilever chip 1030 by actuator 1040 iscontrolled by excitation electronics that include at least two frequencysynthesizers 1080 and 1090. There could be additional synthesizers ifmore than two cantilever eigenmodes are to be employed. The signals fromthese frequency synthesizers could be summed together by an analogcircuit element 1100 or, preferably, a digital circuit element thatperforms the same function. The two frequency synthesizers 1080 and 1090provide reference signals to lockin amplifiers 1110 and 1120,respectively. In the case where more than two eigenmodes are to beemployed, the number of lockin amplifiers will also be increased. Aswith other electronic components in this apparatus, the lockinamplifiers 1110 and 1120 can be made with analog circuitry or withdigital circuitry or a hybrid of both. For a digital lockin amplifier,one interesting and attractive feature is that the lockin analysis canbe performed on the same data stream for both eigenmodes. This impliesthat the same position sensitive detector and analog to digitalconverter can be used to extract information at the two distincteigenmodes.

The lockin amplifiers could also be replaced with rms measurementcircuitry where the rms amplitude of the cantilever oscillation is usedas a feedback signal.

There are a number of variations in the FIG. 1 apparatus that a personskilled in the art could use to extract information relative to thedifferent eigenmodes employed in the present invention. Preferably, adirect digital synthesizer (DDS) could be used to create sine and cosinequadrature pairs of oscillating voltages, each at a frequency matched tothe eigenmodes of the cantilever probe 1030 that are of interest. Thisimplementation also allows dc voltages to be applied, allowing methodssuch as scanning Kelvin probing or simultaneous current measurementsbetween the tip and the sample. The amplitude and phase of eacheigenmode can be measured and used in a feedback loop calculated by thecontroller 1130 or simply reported to the user interface 1140 where itis displayed, stored and/or processed further in an off-line manner.Instead of, or in addition to, the amplitude and phase of the cantilevermotion, the quadrature pairs, usually designated x and y, can becalculated and used in a manner similar to the amplitude and phase.

In one method of using the FIG. 1 apparatus, the cantilever is driven ator near two or more resonances by the single “shake” piezo 1040.Operating in a manner similar to AC mode where the cantilever amplitudeis maintained constant and used as a feedback signal, but employing theteachings of the present invention, there are now a number of choicesfor the feedback loop. Although the work here will focus on using theamplitude of the fundamental (A0), we were able to successfully imageusing one of the higher mode amplitudes (Ai) as a feedback signal aswell as a sum of all the amplitudes A0+A1+ . . . One can also choose toexclude one or more modes from such a sum. So for example, where threemodes are employed, the sum of the first and second could be used tooperate the feedback loop and the third could be used as a carry alongsignal.

Because higher eigenmodes have a significantly higher dynamic stiffness,the energy of these modes can be much larger that that of lowereigenmodes.

The method may be used to operate the apparatus with one flexural modeexperiencing a net attractive force and the other a net repulsive force,as well as operating with each mode experiencing the same net sign offorce, attractive or repulsive. Using this method, with the cantileverexperiencing attractive and repulsive interactions in differenteigenmodes, may provide additional information about sample properties.

One preferred technique for using the aforesaid method is as follows.First, excite the probe tip at or near a resonant frequency of thecantilever keeping the tip sufficiently far from the sample surface thatit oscillates at the free amplitude A10 unaffected by the proximity ofthe cantilever to the sample surface and without making contact with thesample surface. At this stage, the cantilever is not touching thesurface; it turns around before it interacts with significant repulsiveforces.

Second, reduce the relative distance in the Z direction between the baseof the cantilever and the sample surface so that the amplitude of theprobe tip A1 is affected by the proximity of the sample surface withoutthe probe tip making contact with the sample surface. The phase p1 willbe greater than p10, the free first eigenmode phase. This amplitude ismaintained at an essentially constant value during scanning without theprobe tip making contact with the sample surface by setting up afeedback loop that controls the distance between the base of thecantilever and the sample surface.

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode of thecantilever at an amplitude A2. Increase A2 until the second eigenmodephase p2 shows that the cantilever eigenmode is interacting withpredominantly repulsive forces; that is, that p2 is less than p20, thefree second eigenmode phase. This second amplitude A2 is not included inthe feedback loop and is allowed to freely roam over a large range ofvalues. In fact, it is typically better if variations in A2 can be aslarge as possible, ranging from 0 to A20, the free second eigenmodeamplitude.

Fourth, the feedback amplitude and phase, A1 and p1, respectively, aswell as the carry along second eigenmode amplitude and phase, A2 and p2,respectively, should be measured and displayed.

Alternatively, the drive amplitude and/or phase of the second frequencycan be continually adjusted to maintain the second amplitude and/orphase at an essentially constant value. In this case, it is useful todisplay and record the drive amplitude and/or frequency required tomaintain the second amplitude and/or phase at an essentially constantvalue.

A second preferred technique for using the aforesaid method follows thefirst two steps of first preferred technique just described and thencontinues with the following two steps:

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode (orharmonic) of the cantilever at an amplitude A2. Increase A2 until thesecond eigenmode phase p2 shows that the cantilever eigenmode isinteracting with predominantly repulsive forces; that is, that p2 isless than p20, the free second eigenmode phase. At this point, thesecond eigenmode amplitude A2 should be adjusted so that the firsteigenmode phase p1 becomes predominantly less than p10, the free firsteigenmode phase. In this case, the adjustment of the second eigenmodeamplitude A2 has induced the first eigenmode of the cantilever tointeract with the surface in a repulsive manner. As with the firstpreferred technique, the second eigenmode amplitude A2 is not used inthe tip-surface distance feedback loop and should be allowed rangewidely over many values.

Fourth, the feedback amplitude and phase, A1 and p1, respectively, aswell as the carry along second eigenmode amplitude and phase, A2 and p2,respectively, should be measured and displayed.

Either of the preferred techniques just described could be performed ina second method of using the FIG. 1 apparatus where the phase of theoscillating cantilever is used in a feedback loop and the oscillationfrequency is varied to maintain phase essentially constant. In thiscase, it is preferable to use the oscillation frequency as an input intoa z-feedback loop that controls the cantilever-sample separation.

Relative changes in various parameters such as the amplitude and phaseor in-phase and quadrature components of the cantilever at thesedifferent frequencies could also be used to extract information aboutthe sample properties.

A third preferred technique for using the first method of using the FIG.1 apparatus provides an alternative for conventional operation in arepulsive mode, that is where the tip is experiencing a net repulsiveforce. The conventional approach for so operating would be to use alarge amplitude in combination with a lower setpoint, and a cantileverwith a very sharp tip. Using this third preferred technique, however,the operator begins, just as with the first two techniques, by choosingan amplitude and setpoint for the fundamental eigenmode that is smallenough to guarantee that the cantilever is experiencing attractiveforces, that is, that the cantilever is in non-contact mode. As notedbefore, this operational mode can be identified by observing the phaseof the cantilever oscillation. In the non-contact case, the phase shiftis positive, implying that the resonant frequency has been lowered. Withthese conditions on the first eigenmode, the second eigenmode excitationcan be introduced and the amplitude, drive frequency and, if applicable,set point chosen with the following considerations in mind:

1. Both eigenmodes are in the attractive mode, that is to say that thephase shift of both modes is positive, implying both eigenmodefrequencies have been shifted negatively by the tip-sample interactions.Generally, this requires a small amplitude for the second eigenmode.

2. The fundamental eigenmode remains attractive while the secondeigenmode is in a state where the tip-sample interactions cause it to bein both the attractive and the repulsive modes as it is positionedrelative to the surface.

3. The fundamental eigenmode is in an attractive mode and the secondeiegenmode is in a repulsive mode.

4. In the absence of any second mode excitation, the first eigenmode isinteracting with the surface in the attractive mode. After the secondeigenmode is excited, the first eigenmode is in a repulsive mode. Thischange is induced by the addition of the second eigenmode energy. Thesecond eigenmode is in a state where the tip-sample interactions causeit to be attractive and/or repulsive.

5. The first eigenmode is in a repulsive mode and the second mode is ina repulsive mode.

The transition from attractive to repulsive mode in the first eigenmode,as induced by the second eigenmode excitation, is illustrated in FIG. 4,where the amplitude and phase of the first and second eigenmodes areplotted as a function of the distance between the base of the cantileverand the surface of the sample. The point where the tip begins tointeract significantly with the surface is indicated with a solid line4000. The fundamental amplitude 4010 of the cantilever decreases as thecantilever starts to interact with the surface, denoted by the solidline 4000. The associated phase 4020 shows a positive shift, consistentwith overall attractive interactions. For these curves, the secondeigenmode amplitude is zero and therefore not plotted in the Figure (andneither is phase, for the same reason). Next, the second eigenmode isexcited and the same curves are re-measured, together with the amplitudeand phase of the second eigenmode, 4030 and 4040. There is a notablechange in the fundamental eigenmode amplitude 4050 and more strikingly,the fundamental eigenmode phase 4060. The fundamental phase in factshows a brief positive excursion, but then transitions to a negativephase shift, indicating an overall repulsive interaction between the tipand sample. The free amplitude of the first eigenmode is identical inboth cases, the only difference in the measurement being the addition ofenergy exciting the higher oscillatory eigenmode. This excitation issufficient to drive the fundamental eigenmode into repulsiveinteractions with the sample surface. Furthermore, the phase curve ofthe second eigenmode indicates that it is also interacting overallrepulsively with the sample surface.

More complicated feedback schemes can also be envisioned. For example,one of the eigenmode signals can be used for topographical feedbackwhile the other signals could be used in other feedback loops. Anexample would be that A1 is used to control the tip-sample separationwhile a separate feedback loop was used to keep A2 at an essentiallyconstant value rather than allowing it to range freely over many values.A similar feedback loop could be used to keep the phase of the secondfrequency drive p2 at a predetermined value with or without the feedbackloop on A2 being implemented.

As another example of yet another type of feedback that could be used,Q-control can also be used in connection with any of the techniques forusing the first method of using the FIG. 1 apparatus. Using Q-control onany or all of the eigenmodes employed can enhance their sensitivity tothe tip-sample forces and therefore mechanical or other properties ofthe sample. It can also be used to change the response time of theeigenmodes employed which may be advantageous for more rapidly imaging asample. For example, the value of Q for one eigenmode could be increasedand the value for another decreased. This may enhance the result ofmixed attractive/repulsive mode imaging because it is generally easierto keep one eignmode interacting with the sample in repulsive mode witha reduced Q-value or, conversely, in attractive mode with an enhancedQ-value. By reducing the Q-value of the lowest eigenmode and enhancingthe Q-value of the next eigenmode, it is possible to encourage the mixedmode operation of the cantilever; the zeroth eigenmode will be inrepulsive mode while the first eigenmode will more likely remain inattractive mode. Q-control can be implemented using analog, digital orhybrid analog-digital electronics. It can be accomplished using anintegrated control system such as that in the Asylum ResearchCorporation MFP-3D Controller or by after-market modules such as thenanoAnalytics Q-box.

In addition to driving the cantilever at or near more than oneeigenmode, it is possible to also excite the cantilever at or near oneor more harmonics and/or one or more eigenmodes. It has been known forsome time that nonlinear interactions between the tip and the sample cantransfer energy into cantilever harmonics. In some cases this energytransfer can be large but it is usually quite small, on the order of apercent of less of the energy in the eigenmode. Because of this, theamplitude of motion at a harmonic, even in the presence of significantnonlinear coupling is usually quite small. Using the methods describedhere, it is possible to enhance the contrast of these harmonics bydirectly driving the cantilever at the frequency of the harmonic. Tofurther enhance the contrast of this imaging technique it is useful toadjust the phase of the higher frequency drive relative to that of thelower. This method improves the contrast of both conventionalcantilevers and the specially engineered “harmonic” cantileversdescribed by Sahin et al and other researchers.

On many samples, the results of imaging with the present invention aresimilar to, and in some cases superior to, the results of conventionalphase imaging. However, while phase imaging often requires a judiciouschoice of setpoint and drive amplitude to maximize the phase contrast,the method of the present invention exhibits high contrast over a muchwider range of imaging parameters. Moreover, the method also works influid and vacuum, as well as air and the higher flexural modes showunexpected and intriguing contrast in those environments, even onsamples such as DNA and cells that have been imaged numerous timesbefore using more conventional techniques.

Although there is a wide range of operating parameters that yieldinteresting and useful data, there are situations where more carefultuning of the operational parameters will yield enhanced results. Someof these are discussed below. Of particular interest can be regions inset point and drive amplitude space where there is a transition fromattractive to repulsive (or vice versa) interactions in one or more ofthe cantilever eigenmodes or harmonics.

The superior results of imaging with the present invention may be seenfrom an inspection of the images. An example is shown in FIG. 5. Forthis example, the FIG. 1 apparatus was operated using the fundamentaleigenmode amplitude as the error signal and the second eigenmode as acarry-along signal. The topography image 5010 in FIG. 5 shows collagenfibers on a glass surface, an image typical of results with conventionalAC mode from similar samples. The fundamental eigenmode amplitude image5020 is relatively similar, consistent with the fundamental eigenmodeamplitude being used in the feedback loop. The fundamental eigenmodephase channel image 5030 shows some contrast corresponding to edges inthe topography image. This is consistent with the interaction being moreattractive at these regions, again to be expected from surface energyconsiderations (larger areas in proximity will have larger long-rangeattractive forces). Since the fundamental eigenmode amplitude is beingheld relatively constant and there is a relationship between theamplitude and phase, the phase will be constrained, subject to energybalance and the feedback loop that is operating to keep the amplitudeconstant. The second eigenmode amplitude image 5040 shows contrast thatis similar to the fundamental eigenmode phase image 5030. However, thereare some differences, especially over regions thought to be contaminants5041 and 5042. Finally, the second eigenmode phase image 5050 shows themost surprisingly large amount of contrast. The background substrate5053 shows a bright, positive phase contrast. The putative contaminantpatches, 5041, 5042 and 5051 show strikingly dark, negative phase shiftcontrast. Finally, regions where the collagen fibers are suspended 5052show dark, negative phase contrast. In these last regions, the suspendedcollagen fibers are presumably absorbing some of the vibrational energyof the second eigenmode amplitude and thus, changing the response.

When an AFM is operated in conventional amplitude modulated (AM) AC modewith phase detection, the cantilever amplitude is maintained constantand used as a feedback signal. Accordingly, the values of the signalused in the loop are constrained not only by energy balance but also bythe feedback loop itself. Furthermore, if the amplitude of thecantilever is constrained, the phase will also be constrained, subjectto conditions discussed below. In conventional AC mode it is not unusualfor the amplitude to vary by a very small amount, depending on the gainsof the loop. This means that, even if there are mechanical properties ofthe sample that might lead to increased dissipation or a frequency shiftof the cantilever, the z-feedback loop in part corrects for thesechanges and thus in this sense, avoids presenting them to the user.

If the technique for using the present invention involves a mode that isexcited but not used in the feedback loop, there will be no explicitconstraints on the behavior of this mode. Instead it will range freelyover many values of the amplitude and phase, constrained only by energybalance. That is to say, the energy that is used to excite thecantilever motion must be balanced by the energy lost to the tip-sampleinteractions and the intrinsic dissipation of the cantilever. This mayexplain the enhanced contrast we observe in images generated with thetechniques of the present invention.

FIG. 6 demonstrates this idea more explicitly. The first image 6010 isan image of the number of pixels at different amplitudes (horizontalaxis) and phases (vertical axis) in the fundamental eigenmode data forthe collagen sample of FIG. 5. As expected, the amplitude values areconstrained to a narrow range around ˜0.6Amax by the z-feedback loop.Constraining the amplitude values in turn, limits the values that thephase can take to the narrow range around 25°. Thus, when the pixelcounts are plotted, there is a bright spot 6020 with only smallvariations. Small variations in turn imply limited contrast. The secondimage 6030 plots the number of pixels at different amplitudes and phasesin the second eigenmode data for the collagen sample. Since theamplitude of this eigenmode was not constrained by a feedback loop, itvaries from Amax to close to zero. Similarly, the phase ranges over manyvalues. This freedom allows greatly increased contrast in the secondeigenmode images.

The present invention may also be used in apparatus that induce motionin the cantilever other than through a piezoelectric actuator. Thesecould include direct electric driving of the cantilever (“activecantilevers”), magnetic actuation schemes, ultrasonic excitations,scanning Kelvin probe and electrostatic actuation schemes.

Direct electric driving of the cantilever (“active cantilevers”) usingthe present invention has several advantages over conventional piezoforce microscopy (PFM) where the cantilever is generally scanned overthe sample in contact mode and the cantilever voltage is modulated in amanner to excite motion in the sample which in turn causes thecantilever to oscillate.

FIG. 2 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with an active cantilever. This apparatushas similarities to that shown in FIG. 1, as well as differences. In theFIG. 2 apparatus, like the FIG. 1 apparatus, one of the frequencysources 1080 is used to excite motion of the cantilever probe 1020through a mechanical actuator 1040, preferably a piezoelectric actuator,but other methods to induce cantilever motion known to those versed inthe art could also be used, which drives the chip 1030 of the cantileverprobe 1020, However, in the FIG. 2 apparatus, the frequency source 1080communicates directly 2010 with the actuator 1040 instead of beingsummed together with the second frequency source 1090, as in the FIG. 1apparatus. The second frequency source 1090 in the FIG. 2 apparatus isused to vary the potential of the cantilever probe 1020 which in turncauses the sample 1010 to excite motion in the cantilever probe 1020 ata different eigenmode than that excited by the first frequency source1080. The resulting motion of the cantilever probe 1020 interacting withthe sample 1010 will contain information on the sample topography andother properties at the eigenmode excited by the first frequency source1080 and information regarding the voltage dependent properties of thesample at the eigenmode excited by the second frequency source 1090. Thesample holder 2030 can optionally be held at a potential, or at ground,to enhance the effect.

In one method of using the FIG. 2 apparatus, the amplitude of thecantilever at the frequency of the first source 1080 is used as theerror signal. The amplitude and phase (or in-phase and quadraturecomponents) at the frequency of the second source 1090 or a harmonicthereof will contain information about the motion of the sample andtherefore the voltage dependent properties of the sample. One example ofthese properties is the piezo-response of the sample. Another is theelectrical conductivity, charge or other properties that can result inlong range electrostatic forces between the tip and the sample.

FIG. 3 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with the second frequency source modulatinga magnetic field that changes a property of the sample. In the FIG. 3apparatus, the situation with the first frequency source 1080 isidentical to the situation in the FIG. 2 apparatus. However, instead ofthe second frequency source 1090 being used to vary the potential of thecantilever probe 1020, as with the FIG. 2 apparatus, in the FIG. 3apparatus the second frequency source 1090 modulates the current throughan excitation coil 3010 which in turn modulates the magnetic state of amagnetic circuit element 3020. Magnetic circuit element 3020 could beused to modulate the field near an active sample or the excitation coil3010. Alternatively, magnetic circuit element 3020 could comprise thesample, as in the case of a magnetic recording head.

The FIG. 3 apparatus can be used with any other sort of ‘active’ samplewhere the interaction between the cantilever and the sample can bemodulated at or near one or more of the cantilever flexural resonancesby one of the frequency sources 1080 or 1090. This could also beextended to high frequency measurements such as described in Proksch etal., Appl. Phys. Lett., vol. (1999). Instead of the modulation describedin that paper, the envelope of the high frequency carrier could bedriven with a harmonic of one or more flexural resonances. This methodof measuring signals other than topographic has the advantage ofrequiring only one pass to complete as opposed to “LiftMode” or Nap modethat require temporally separated measurements of the topographic andother signals.

Another example of a preferred embodiment of an apparatus and method forusing the present invention is from the field of ultrasonic forcemicroscopy. In this embodiment, one or more eigenmodes are used for thez-feedback loop and one or more additional eigenmodes can be used tomeasure the high frequency properties of the sample. The high frequencycarrier is amplitude modulated and either used to drive the sampledirectly or to drive it using the cantilever as a waveguide. Thecantilever deflection provides a rectified measure of the sampleresponse at the carrier frequency.

Another group of embodiments for the present invention has similaritiesto the conventional force modulation technique described in theBackground to the Invention and conventional PFM where the cantilever isscanned over the sample in contact mode and a varying voltage is appliedto the cantilever. In general this group may be described as contactresonance embodiments. However, these embodiments, like the otherembodiments already described, make use of multiple excitation signals.

FIG. 8 is a block diagram of the first of these embodiments, which maybe referred to as Dual Frequency Resonance Tracking Piezo ForceMicroscopy (DFRT PFM). In the DFRT PFM apparatus of FIG. 8 thecantilever probe 1020 is positioned above a sample 1010 withpiezoelectric properties and scanned relative to the sample 1010 by ascanning apparatus 1070 using contact mode. Unlike conventional contactmode however the chip 1030 of the cantilever probe 1020, or thecantilever probe 1020 itself (alternative not shown), is driven byexcitation electronics that include at least two frequency synthesizers1080 and 1090. The cantilever probe 1020 responds to this excitation bybuckling up and down much as a plucked guitar string. The signals fromthese frequency synthesizers could be summed together by an analogcircuit element 1100 or, preferably, a digital circuit element thatperforms the same function. The two frequency synthesizers 1080 and 1090provide reference signals to lockin amplifiers 1110 and 1120,respectively. The motion of the cantilever probe 1020 relative to theframe of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved vertically relative to the sample1010, in order to maintain constant force, by a scanning apparatus 1070,preferably a piezo/flexure combination, but other methods known to thoseversed in the art could also be used. The amplitude and phase of eachfrequency at which the cantilever probe 1020 is excited can be measuredand used in a feedback loop calculated by the controller 1130 or simplyreported to the user interface 1140 where it is displayed, stored and/orprocessed further in an off-line manner. Instead of, or in addition to,the amplitude and phase of the cantilever motion, the quadrature pairs,usually designated x and y, can be calculated and used in a mannersimilar to the amplitude and phase.

In one method of using the FIG. 8 apparatus, the topography of thesample would be measured in contact mode while the amplitude and phaseof the cantilever probe 1020 response to the applied potential at thelowest contact resonance and at the next highest contact resonance issimultaneously measured. The responses can be analyzed to determinewhether they originate from the actual piezoelectric response of thesample or from crosstalk between the topography and any electric forcesbetween the tip of the cantilever probe 1020 and the sample. Even moreinformation can be obtained if more frequencies are utilized.

FIG. 12 shows three examples of the changes in the native phase 12015and amplitude 12010 of a cantilever with a resonant frequency f0 causedby interactions between the tip and the sample using DFRT PFM methods.These examples are a subset of changes that can be observed. In thefirst example, the resonant frequency is significantly lowered to f0′but not damped. The phase 12085 and amplitude 12080 change but littlerelative to the native phase 12015 and amplitude 12010. In the secondexample the resonant frequency is again lowered to f0′, this time withdamping of the amplitude. Here the phase 12095 is widened and theamplitude 12090 is appreciably flattened. Finally, in the third example,the resonant frequency is again dropped to f0′, this time with areduction in the response amplitude. This yields a phase curve with anoffset 12105 but with the same width as the second case 12095 and areduced amplitude curve 12100 with the damping equivalent to that of thesecond example. If there is an offset in the phase versus frequencycurve as there is in this third example, prior art phase locked-loopelectronics will not maintain stable operation. For example, if thephase set-point was made to be 90 degrees, it would never be possible tofind a frequency in curve 12105 where this condition was met. Oneexample of these things occurring in a practical situation is in DRFTPFM when the tip crosses from an electric domain with one orientation toa second domain with another orientation. The response induced by thesecond domain will typically have a phase offset with respect to thefirst. This is, in fact where the large contrast in DFRT PFM phasesignals originates.

FIG. 9 shows the cantilever response when the applied potential is sweptfrom dc to 2 MHz using the DFRT PFM apparatus. Three resonance peaks arevisible. Depending on the cantilever probe and the details of thetip-sample contact mechanics, the number, magnitude, breadth andfrequency of the peaks is subject to change. Sweeps such as these areuseful in choosing the operating points for imaging and othermeasurements. In a practical experiment, any or all of these resonancepeaks or the frequencies in between could be exploited by the methodssuggested above.

FIG. 19 shows a measurement that can be made using DFRT PFM techniques.A phase image 19010 shows ferroelectric domains written onto a sol-gelPZT surface. Because of the excellent separation between topography andPFM response possible with DFRT PFM, the phase image shows only piezoresponse, there is no topographic roughness coupling into the phase. Thewritten domains appear as bright regions. The writing was accomplishedby locally performing and measuring hysteresis loops by applying a dcbias to the tip during normal DFRT PFM operation. This allows the localswitching fields to be measured. The piezo phase 19030 and the amplitude19040 during a measurement made at location 19020 are plotted as afunction of the applied dc bias voltage. The loops were made followingStephen Jesse et al, Rev. Sci. Inst. 77, 073702 (2006). Other loops weretaken at the bright locations in image 19010, but are not shown in theFigure.

DFRT PFM is very stable over time in contrast to single frequencytechniques. This allows time dependent processes to be studied as isdemonstrated by the sequence of images, 19010, 19050, 19060, 19070 and19080 taken over the span of about 1.5 hours. In these images, thewritten domains are clearly shrinking over time.

In AC mode atomic force microscopy, relatively tiny tip-sampleinteractions can cause the motion of a cantilever probe oscillating atresonance to change, and with it the resonant frequency, phase,amplitude and deflection of the probe. Those changes of course are thebasis of the inferences that make AC mode so useful. With contactresonance techniques the contact between the tip and the sample also cancause the resonant frequency, phase and amplitude of the cantileverprobe to change dramatically.

The resonant frequency of the cantilever probe using contact resonancetechniques depends on the properties of the contact, particularly thecontact stiffness. Contact stiffness in turn is a function of the localmechanical properties of the tip and sample and the contact area. Ingeneral, all other mechanical properties being equal, increasing thecontact stiffness by increasing the contact area, will increase theresonant frequency of the oscillating cantilever probe. Thisinterdependence of the resonant properties of the oscillating cantileverprobe and the contact area represents a significant shortcoming ofcontact resonance techniques. It results in “topographical crosstalk”that leads to significant interpretational issues. For example, it isdifficult to know whether or not a phase or amplitude change of theprobe is due to some sample property of interest or simply to a changein the contact area.

The apparatus used in contact resonance techniques can also cause theresonant frequency, phase and amplitude of the cantilever probe tochange unpredictably. Examples are discussed by Rabe et al., Rev. Sci.Instr. 67, 3281 (1996) and others since then. One of the most difficultissues is that the means for holding the sample and the cantilever probeinvolve mechanical devices with complicated, frequency dependentresponses. Since these devices have their own resonances and damping,which are only rarely associated with the sample and tip interaction,they may cause artifacts in the data produced by the apparatus. Forexample, phase and amplitude shifts caused by the spurious instrumentalresonances may freely mix with the resonance and amplitude shifts thatoriginate with tip-sample interactions.

It is advantageous to track more than two resonant frequencies as theprobe scans over the surface when using contact resonance techniques.Increasing the number of frequencies tracked provides more informationand makes it possible to over-constrain the determination of variousphysical properties. As is well known in the art, this is advantageoussince multiple measurements will allow better determination of parametervalues and provide an estimation of errors.

Since the phase of the cantilever response is not a well behavedquantity for feedback purposes in PFM, we have developed other methodsfor measuring and/or tracking shifts in the resonant frequency of theprobe. One method is based on making amplitude measurements at more thanone frequency, both of which are at or near a resonant frequency. FIG.15 illustrates the idea. The original resonant frequency curve 14010 hasamplitudes A1 14030 and A2 14020, respectively, at the two drivefrequencies f1 and f2. However, if the resonant frequency shifted to alower value, the curve shifts to 14050 and the amplitudes at themeasurement frequencies change, A′1 14035 increasing and A′2 14025decreasing. If the resonant frequency were higher, the situation wouldreverse, that is the amplitude A′1 at drive frequency f1 would decreaseand A′2 at f2 would increase.

There are many methods to track the resonant frequency with informationon the response at more than one frequency. One method with DFRT PFM isto define an error signal that is the difference between the amplitudeat f1 and the amplitude at f2, that is A1 minus A2. A simpler examplewould be to run the feedback loop such that A1 minus A2=0, althoughother values could equally well be chosen. Alternatively both f1 and f2could be adjusted so that the error signal, the difference in theamplitudes, is maintained. The average of these frequencies (or evensimply one of them) provides the user with a measure of the contactresonant frequency and therefore the local contact stiffness. It is alsopossible to measure the damping and drive with the two values ofamplitude. When the resonant frequency has been tracked properly, thepeak amplitude is directly related to the amplitude on either side ofresonance. One convenient way to monitor this is to simply look at thesum of the two amplitudes. This provides a better signal to noisemeasurement than does only one of the amplitude measurements. Other,more complicated feedback loops could also be used to track the resonantfrequency. Examples include more complex functions of the measuredamplitudes, phases (or equivalently, the in-phase and quadraturecomponents), cantilever deflection or lateral and/or torsional motion.

The values of the two amplitudes also allow conclusions to be drawnabout damping and drive amplitudes. For example, in the case of constantdamping, an increase in the sum of the two amplitudes indicates anincrease in the drive amplitude while the difference indicates a shiftin the resonant frequency.

Finally, it is possible to modulate the drive amplitude and/orfrequencies and/or phases of one or more of the frequencies. Theresponse is used to decode the resonant frequency and, optionally,adjust it to follow changes induced by the tip-sample interactions.

FIG. 10 shows the results of a measurement of a piezoelectric materialusing DFRT PFM methods. Contact mode is used to image the sampletopography 10010 and contact resonance techniques used to image thefirst frequency amplitude 10020, the second frequency amplitude 10030,the first frequency phase 10040 and the second frequency phase 10050. Inthis experiment, the two frequencies were chosen to be close to thefirst contact resonance, at roughly the half-maximum point, with thefirst frequency on the lower side of the resonance curve and the secondon the upper side. This arrangement allowed some of the effects ofcrosstalk to be examined and potentially eliminated in subsequentimaging.

Another multiple frequency technique is depicted in FIG. 2, an apparatusfor using the present invention with a conductive (or active)cantilever, and the methods for its use may also be advantageous inexamining the effects of crosstalk with a view to potentiallyeliminating them in subsequent imaging. For this purpose the inventorsrefer to this apparatus and method as Dual Frequency Piezo ForceMicroscopy (DF PFM). In the DF PFM apparatus of FIG. 2 the response todriving the tip voltage of the cantilever probe, due to thepiezoelectric action acting through the contact mechanics, willtypically change as the probe is scanned over the surface. The firstsignal will then be representative of changes in the contact mechanicsbetween the tip and sample. The second signal will depend both oncontact mechanics and on the piezo electrical forces induced by thesecond excitation signal between the tip and sample. Differences betweenthe response to the first excitation and the response to the second arethus indicative of piezoelectric properties of the sample and allow thecontact mechanics to be separated from such properties.

As noted, the user often does not have independent knowledge about thedrive or damping in contact resonance. Furthermore, models may be oflimited help because they too require information not readily available.In the simple harmonic oscillator model for example, the drive amplitudeA_(drive), drive phase φ_(drive), resonant frequency ω₀ and qualityfactor Q (representative of the damping) will all vary as a function ofthe lateral tip position over the sample and may also vary in timedepending on cantilever mounting schemes or other instrumental factors.In conventional PFM, only two time averaged quantities are measured, theamplitude and the phase of the cantilever (or equivalently, the in-phaseand quadrature components). However, in dual or multiple frequencyexcitations, more measurements may be made, and this will allowadditional parameters to be extracted. In the context of the SHO model,by measuring the response at two frequencies at or near a particularresonance, it is possible to extract four model parameters. When the twofrequencies are on either side of resonance, as in the case of DFRT PFMfor example, the difference in the amplitudes provides a measure of theresonant frequency, the sum of the amplitudes provides a measure of thedrive amplitude and damping of the tip-sample interaction (or qualityfactor), the difference in the phase values provides a measure of thequality factor and the sum of the phases provides a measure of thetip-sample drive phase.

Simply put, with measurements at two different frequencies, we measurefour time averaged quantities, A₁, A₂, φ₁, φ₂ that allow us to solve forthe four unknown parameters A_(drive), φ_(drive), f₀ and Q.

FIG. 18 illustrates the usefulness of measuring the phase as a means ofseparating changes in the quality factor Q from changes in the driveamplitude A_(drive). Curve 18010 shows the amplitude response of anoscillator with a resonance frequency of f₀=320 kHz, a quality factorQ=110 and a drive amplitude A_(drive)=0.06 nm. Using DFRT PFMtechniques, the amplitude A₁ 18012 is measured at a drive frequency f₁and the amplitude A₂ 18014 is measured at a drive frequency f₂. Curve18030 shows what happens when the Q value increases to 150. The firstamplitude A₁ 18032 increases because of this increase in Q, as does thesecond amplitude A₂ 18034. Curve 18050 shows what happens when thequality factor Q, remains at 110 and the drive amplitude A_(drive)increases from 0.06 nm to 0.09 nm. Now, the amplitude measurements madeat f₁ 18052 and f₂ 18054 are exactly the same as in the case where the Qvalue increased to 150, 18032 and 18034, respectively. The amplituderesponse does not separate the difference between increasing the Q valueor increasing the drive amplitude A_(drive).

This difficulty is surmounted by measuring the phase. Curves 18020,18040 and 18060 are the phase curves corresponding to the amplitudecurves 18010, 18030 and 18050 respectively. As with the amplitudemeasurements, the phase is measured at discrete frequency values, f₁ andf₂. The phase curve 18020 remains unchanged 18060 when the driveamplitude Adrive increases from 0.06 nm to 0.09 nm. Note that the phasemeasurements 18022 and 18062 at f₁ for the curves reflecting an increasein drive amplitude but with the same quality factor are the same, as arethe phase measurements 18024 and 18064 at f₂. However, when the qualityfactor Q increases, the f₁ phase 18042 decreases and the f₂ phase 18044increases. These changes clearly separate drive amplitude changes from Qvalue changes.

In the case where the phase baseline does not change, it is possible toobtain the Q value from one of the phase measurements. However, as inthe case of PFM and thermal modulated microscopy, the phase baseline maywell change. In this case, it is advantageous to look at the differencein the two phase values. When the Q increases, this difference 18080will also increase. When the Q is unchanged, this difference 18070 isalso unchanged.

If we increase the number of frequencies beyond two, other parameterscan be evaluated such as the linearity of the response or the validityof the simple harmonic oscillator model

Once the amplitude, phase, quadrature or in-phase component is measuredat more than one frequency, there are numerous deductions that can bemade about the mechanical response of the cantilever to various forces.These deductions can be made based around a model, such as the simpleharmonic oscillator model or extended, continuous models of thecantilever or other sensor. The deductions can also be made using apurely phenomenological approach. One simple example in measuringpassive mechanical properties is that an overall change in theintegrated amplitude of the cantilever response, the response of therelevant sensor, implies a change in the damping of the sensor. Incontrast, a shift in the “center” of the amplitude in amplitude versusfrequency measurements implies that the conservative interactionsbetween the sensor and the sample have changed.

This idea can be extended to more and more frequencies for a betterestimate of the resonant behavior. It will be apparent to those skilledin the art that this represents one manner of providing a spectrum ofthe sensor response over a certain frequency range. The spectralanalysis can be either scalar or vector. This analysis has the advantagethat the speed of these measurements is quite high with respect to otherfrequency dependent excitations.

In measuring the frequency response of a sensor, it is not required toexcite the sensor with a constant, continuous signal. Other alternativessuch as so-called band excitation, pulsed excitations and others couldbe used. The only requirement is that the appropriate reference signalbe supplied to the detection means.

FIG. 16 shows one embodiment of a multi-frequency approach, with eightfrequencies f₁ through f₈ being driven. As the resonance curve changesin response to tip-surface interactions, a more complete map of thefrequency response is traced out. This may be particularly useful whenmeasuring non-linear interactions between the tip and the sample becausein that case the simple harmonic oscillator model no longer applies. Theamplitude and phase characteristics of the sensor may be significantlymore complex. As an example of this sort of measurement, one can drivethe cantilever at one or more frequencies near resonance and measure theresponse at nearby frequencies.

Scanning ion conductance microscopy, scanning electrochemicalmicroscopy, scanning tunneling microscopy, scanning spreading resistancemicroscopy and current sensitive atomic force microscopy are allexamples of localized transport measurements that make use ofalternating signals, again sometimes with an applied dc bias. Electricalforce microscopy, Kelvin probe microscopy and scanning capacitancemicroscopy are other examples of measurement modes that make use ofalternating signals, sometimes with an applied dc bias. These and othertechniques known in the art can benefit greatly from excitation at morethan one frequency. Furthermore, it can also be beneficial if excitationof a mechanical parameter at one or more frequencies is combined withelectrical excitation at the same or other frequencies. The responsesdue to these various excitations can also be used in feedback loops, asis the case with Kelvin force microscopy where there is typically afeedback loop operating between a mechanical parameter of the cantileverdynamics and the tip-sample potential.

Perhaps the most popular of the AC modes is amplitude-modulated (AM)Atomic Force Microscopy (AFM), sometimes called (by Bruker Instruments)tapping mode or intermittent contact mode. Under the name “tapping mode”this AC mode was first coined by Finlan, independently discovered byGleyzes, and later commercialized by Digital Instruments.

AM AFM imaging combined with imaging of the phase, that is comparing thesignal from the cantilever oscillation to the signal from the actuatordriving the cantilever and using the difference to generate an image, isa proven, reliable and gentle imaging/measurement method with widespreadapplications. The first phase images (of a wood pulp sample) werepresented at a meeting of Microscopy and Microanalysis. Since then,phase imaging has become a mainstay in a number of AFM applicationareas, most notably in polymers where the phase channel is often capableof resolving fine structural details.

The phase response has been interpreted in terms of the mechanical andchemical properties of the sample surface. Progress has been made inquantifying energy dissipation and storage between the tip and samplewhich can be linked to specific material properties. Even with theseadvances, obtaining quantitative material or chemical properties remainsproblematic. Furthermore, with the exception of relatively soft metalssuch as In-Tn solder, phase contrast imaging has been generally limitedto softer polymeric materials, rubbers, fibrous natural materials. Onthe face of it this is somewhat puzzling since the elastic and lossmoduli of harder materials can vary over many orders of magnitude.

The present invention adapts techniques used recently in research onpolymers, referred to there as loss tangent imaging, to overcome some ofthese difficulties. Loss tangent imaging recasts our understanding ofphase imaging by linking energy dissipation and energy storage into oneterm that includes both the dissipated and the stored energy of theinteraction between the tip and the sample. The linkage becomes afundamental material property—if for example the dissipation increasesbecause of an increase in the indentation depth, the stored elasticenergy will also increase. In the case of linear viscoelastic materials,the ratio between the dissipated energy and elastically stored energy isthe loss tangent. This is similar to other dimensionless approaches tocharacterizing loss and storage in materials such as the coefficient ofrestitution. The loss tangent approach to materials has very earlyroots, dating back at least to the work of Zener in 1941. One shouldnote however that many materials are not linear viscoelastic materials,especially in the presence of large strains (>1%). The degree ofdeviation from the behavior of linear viscoelastic materials exhibitedby other materials is in itself useful and interesting to measure.

In addition to loss tangent imaging, the present invention includes thequantitative and high sensitivity of simultaneous operation in afrequency modulated (FM) mode. For this purpose the AFM is set up forbimodal imaging with two feedback loops, the first using the firstresonance of the cantilever and the second another higher resonance. Thefirst loop is an AM mode feedback loop that controls the tip-sampleseparation by keeping the amplitude of the cantilever constant (andproduces a topographic image from the feedback signals) and at the sametime compares the signal from the cantilever oscillation to the signalfrom the actuator driving the cantilever to measure changes in phase asthe tip-sample separation is maintained constant. The second feedbackloop is a FM mode feedback loop that controls the tip-sample separationby varying the drive frequency of the cantilever. The frequency isvaried in FM mode through a phase-locked loop (PLL) that keeps thephase—a comparison of the signal from the cantilever oscillation to thesignal from the actuator driving the cantilever at the secondresonance—at 90 degrees by adjusting the frequency. It is also possibleto implement another feedback loop to keep the amplitude of thecantilever constant through the use of automatic gain control (AGC). Ifthe AGC is implemented, output amplitude is constant. Otherwise, if theamplitude is allowed to vary, it is termed constant excitation mode.

Much of the initial work with FM mode was in air and it has a longtradition of being applied to vacuum AFM studies (including UHV),routinely attaining atomic resolution and even atomic scale chemicalidentification. Recently there has been increasing interest in theapplication of this technique to various samples in liquid environments,particularly biological samples. Furthermore, FM AFM has demonstratedtrue atomic resolution imaging in liquid where the low Q results in areduction in force sensitivity. One significant challenge of FM AFM hasbeen with stabilizing feedback loops.

Briefly, when AM mode imaging with phase is combined with FM modeimaging using bimodal imaging techniques, the topographic feedbackoperates in AM mode while the second resonant mode drive frequency isadjusted to keep the phase at 90 degrees. With this approach, frequencyfeedback on the second resonant mode and topographic feedback on thefirst are decoupled, allowing much more stable, robust operation. The FMimage returns a quantitative value of the frequency shift that in turndepends on the sample stiffness and can be applied to a variety ofphysical models.

Bimodal imaging involves using more than one resonant vibrational modeof the cantilever simultaneously. A number of multifrequency AFM schemeshave been proposed to improve high resolution imaging, contrast andquantitative mapping of material properties, some of which have alreadybeen discussed above.

With bimodal imaging the resonant modes can be treated as independentchannels, with each channel having separate observables, generally theamplitude and phase. The cantilever is driven at two flexuralresonances, typically the first two, as has been described above. Theresponse of the cantilever at the two resonances is measured and used indifferent ways as shown in FIG. 20. It will be noted that the FIG. 20apparatus bear some resemblance to the apparatus shown in FIG. 1.

FIG. 20 is a block diagram of a preferred embodiment of an apparatus forprobing two flexural resonances of a cantilever in accordance with thepresent invention. The sample 1010 is positioned below the cantileverprobe 1020. The chip of the cantilever probe 1030 is driven by amechanical actuator 1040, preferably a piezoelectric actuator, but othermethods to induce cantilever motion known to those versed in the artcould also be used. The motion 1150 of the cantilever 1020 relative tothe frame of the microscope (not shown) is measured with a detector (notshown), which could be an optical lever or another method known to thoseversed in the art. The cantilever chip 1030 is moved relative to thesample 1010 by a scanning apparatus (not shown), preferably apiezo/flexure combination, but other methods known to those versed inthe art could also be used.

The motion imparted to the cantilever chip 1030 by actuator 1040 iscontrolled by excitation electronics that include at least two frequencysynthesizers 1080 and 1090. The signals from these frequencysynthesizers could be summed together by an analog circuit element 1100or, preferably, a digital circuit element that performs the samefunction. The two frequency synthesizers 1080 and 1090 provide referencesignals to lockin amplifiers 1110 and 1120, respectively. As with otherelectronic components in this apparatus, the lockin amplifiers 1110 and1120 can be made with analog circuitry or with digital circuitry or ahybrid of both. For a digital lockin amplifier, one interesting andattractive feature is that the lockin analysis can be performed on thesame data stream for both flexural resonances. This implies that thesame position sensitive detector and analog to digital converter can beused to extract information at the two distinct resonances.

Resonance 1: As shown in the upper shaded area of FIG. 20, the flexuralresonance signal from frequency synthesizer 1080 is compared to thecantilever deflection signal 1150 through lockin amplifier 1110. Thisfeedback loop controls the z actuator (not shown) which moves thecantilever chip 1030 relative to the sample 1010 and thus controls theamplitude of the cantilever 1020 and the tip-sample separation. Theamplitude signal resulting from this feedback is used to create atopographic image of the sample 1010. Simultaneously, the phase of thecantilever 1020 is calculated from this comparison and together with theamplitude signal is used to generate the tip-sample loss tangent image.

Resonance 2: As shown in the lower shaded area of FIG. 20, the flexuralresonance signal from frequency synthesizer 1090 is compared to thecantilever deflection signal 1150 through lockin amplifier 1120 and asecond phase of the cantilever 1020 is calculated from this comparison.The PLL device 1160 in turn maintains this phase at 90 degrees by makingappropriate adjustments in the flexural resonance signal from frequencysynthesizer 1090. The required adjustment provides a FM based measure oftip-sample stiffness and dissipation. Tip-sample stiffness anddissipation can also be measured from the amplitude and phase of theflexural resonance signal from frequency synthesizer 1090. FM mode mayalso employ an AGC device to maintain the amplitude of the cantileverprobe 1020 at a constant value.

The foregoing bimodal imaging approach to quantitative measurements hasthe great advantage of stability when used with Loss Tangent and AM/FMimaging techniques. With topographic feedback confined to the firstresonant mode and FM control to the second resonant mode, even if thePLL or AGC control loops become unstable and oscillate, there is littleor no effect on the ability of the first mode to stably track thesurface topography. As is well known in the art, AM mode AFM imaging,where the topographic feedback is controlled by the oscillations of thefirst mode is extremely robust and stable. Thus, the overall imagingperformance, where topographic and other information are gatheredsimultaneously, is very stable and robust.

In order to highlight some important limitations it is useful to take amathematical approach to Loss Tangent imaging. As already noted in thediscussion of AM AFM operation, the amplitude of the first resonant modeis used to maintain the tip-sample distance. The control voltage,typically applied to a z-actuator, results in a topographic image of thesample surface. At the same time, the phase of the first resonant modewill vary in response to the tip-sample interaction. This phase reflectsboth dissipative and conservative interactions. A tip which indents asurface will both dissipate viscous energy and store elastic energy—thetwo are inextricably linked. The loss tangent can be employed to measurethe tip-sample interaction. As mentioned, the loss tangent is adimensionless parameter which measures the ratio of energy dissipated toenergy stored in a cycle of a periodic deformation. The followingrelation involving the measured cantilever amplitude V and phase φdefines the loss tangent for tip-sample interaction:

$\begin{matrix}{{\tan \; \delta} = {\frac{G^{''}}{G^{\prime}} = {{\frac{\langle{F_{ts} \cdot \overset{.}{z}}\rangle}{\omega {\langle{F_{ts} \cdot z}\rangle}} \approx \frac{\frac{V\mspace{34mu} \omega}{V_{free}\omega_{free}} - {\sin \; \varphi}}{{\cos \; \varphi} - {Q\frac{V}{V_{free}}( {1 - \frac{\omega^{1}}{\omega_{free}^{2}}} )}}} = {\frac{{\Omega \; \alpha} - {\sin \; \varphi}}{{Q\; {\alpha ( {1 - \Omega^{2}} )}} - {\cos \; \varphi}}.}}}} & ({FullTand})\end{matrix}$

In this expression, F_(ts) is the tip-sample interaction force, z is thetip motion, ż is the tip velocity, ω is the angular frequency at whichthe cantilever is driven and

represents a time-average. The parameter V_(free) is the “free” resonantamplitude of the first mode, measured at a reference position. Note thatbecause the amplitudes appear as ratios in Equation (1), they can beeither calibrated or uncalibrated in terms of optical detectorsensitivity. In the final expression in (FullTand) we have defined theratios Ω≡ω/ω_(free) and a α≡A/A_(free)=V/V_(free). If we operate onresonance (Ω=1), the expression can be simplified to:

$\begin{matrix}{{\tan \; \delta} = {\frac{\langle{F_{ts} \cdot \overset{.}{z}}\rangle}{\omega {\langle{F_{ts} \cdot z}\rangle}} \approx {\frac{{\sin \; \varphi} - \alpha}{\cos \; \varphi}.}}} & ({SimpleTand})\end{matrix}$

Equation FullTand differs by a factor of −1 from an earlier version,because it is assumed here that the virial

F_(ts)·z

is positive for repulsive mode operation, which means tan δ>0 inrepulsive mode.

There are some important implications of these equations:

1. Attractive interactions between the tip and the sample will ingeneral make the elastic denominator

F_(ts)·z

of equations FullTand and SimpleTand smaller. This will increase thecantilever loss tangent and therefore over-estimate the sample losstangent.

2. Tip-sample damping with origins other than the sample loss modulus,originating from interactions between, for example, a water layer oneither the tip or the sample will increase the denominator in equationsFullTand and SimpleTand.

These implications point out an important limitation of loss tangentimaging. Equations (FullTand) and (SimpleTand) really represent the losstangent of the cantilever, but not necessarily the loss tangentoriginating from the linear viscoelastic behavior of the samplemechanics: G″ and G′.

Loss tangent analysis has been under-utilized. In the past twenty plusyears of tapping mode imaging, there are many examples of phase imagingof polymeric materials and very few of metals and ceramics with a losstangent less than 0.01, that is, tan δ{tilde under (<)}10⁻². Thereappears to be an impression that less elastic materials are more lossy.However, there are many examples where a stiffer material might alsoexhibit higher dissipation. This underscores the danger in simplyinterpreting phase contrast only in terms of sample elasticity.

Furthermore experimentalists bear the burden of setting up the AFM andits control algorithms such that the loss tangent of interest is beingmeasured. As mentioned above, a number of conservative and dissipativeinteractions contribute to loss tangent estimation in addition to thelinear viscoelastic interactions between the tip and the sample. Infact, depending on the experimental parameters and settings, the sampleloss and storage moduli may contribute only a small portion of thesignal in the loss tangent estimation. For example, when imaging themechanical loss tangent of a polymer surface it is important to operatein repulsive mode, so that the cantilever interacts with the short-rangerepulsive forces controlled by the sample's elastic and loss moduliproperties. In this section we consider some of these contributions,including experimental factors such as air damping and surface hydrationlayers and material effects such as viscoelastic nonlinearity.

Proper choice of the zero-dissipation point is critical for propercalibration of the tip-sample dissipation. In particular, squeeze filmdamping, C. P. Green and J. E. Sader, Frequency response of cantileverbeams immersed in viscous fluids near a solid surface with applicationsto the atomic force microscope, J. Appl. Phys. 98, 114913 (2005); M. Baoand H. Yang, Squeeze film air damping in MEMS, Sens. Actuators, A 136, 3(2007), can have a strong effect on the measured dissipation. Squeezefilm damping causes the cantilever damping to increase as the body ofthe cantilever moves closer to the sample surface. For rough or unevensurfaces, this can mean that the cantilever body height changes withrespect to the average sample position enough to cause crosstalkartifacts in the measured dissipation and therefore the measured losstangent.

An example of squeeze film damping effects is shown in FIG. 21. Thesample there was a silicon (Si) wafer patterned with a film of SU-8, anepoxy resin commonly used for photolithography. The elastic modulus ofsilicon is relatively high (˜150-160 GPa), while that of SU-8 is muchlower (˜2-4 GPa), H. Lorenz, M. Despont, M. Fahrni, N. LaBianca, P.Vettiger and P. Renaud, SU-8: a low-cost negative resist for MEMS, J.Micromech. Microeng. 7, 121 (1997). However, because SU-8 is a polymer,its viscoelastic modulus is expected to be much greater than that of Si.An Olympus AC240 cantilever was used to acquire topography and losstangent images. As the topography image 2101 in FIG. 21 shows, the SU-8film was >1.5 μm thick.

The loss tangent image 2102 in FIG. 21 has a large overall loss tangentvalue of ˜0.3. However, contrary to the expectation, the loss tangentmeasured on the Si is larger than that measured on the SU-8: tanδ_(Si)>tan δ_(SU8). As we will show below, these issues are explained byconsidering another source of energy dissipation—squeeze film damping.Because the Si substrate is ˜1.5 μm lower than the SU-8 film, thecantilever experiences greatly increased squeeze film damping whenmeasuring the Si regions. This is interpreted as a larger loss tangentover the Si regions. Images 2105 and 2106 of FIG. 21 show that thereference free air amplitudes for the cantilever are different over thetwo regions, presumably due to the large difference in height betweenthe two materials. As mentioned above, these reference values arecritical for correctly calculating the loss tangent.

To correct for squeeze film damping effects in loss tangentmeasurements, we use the two-pass imaging technique depicted in FIG. 22.As shown image 2202 of FIG. 22, the first pass of this technique is thenormal AM mode imaging pass, shown in blue. The second pass 2203 of thetechnique is a phase-locked loop (PLL) reference calibration pass, or“nap pass,” shown in red. The cantilever is raised a fixed relativeheight parameter (Δz) above each (x, y) pixel location of the sample andthe same x-y scan as with the first pass is repeated. The color-codedloss tangent equation on the right side of FIG. 22 explains how thetwo-pass procedure is used to make the corrected calibration. Thecantilever drive frequency ω and quality factor Q, both colored green inthe color-coded loss tangent equation on the right side of FIG. 22, aremeasured far away from the surface of the sample in the initialcantilever tune as shown in image 2201 of FIG. 22. The referencefrequency ω_(free) and reference amplitude A_(free) or V_(free) aremeasured by operating the cantilever in a PLL for the first pass. Asalready noted the PLL keeps the frequency and amplitude at referenceduring the second pass.

In order to provide a concrete demonstration of the result of two-passimaging technique, we applied the technique to the Si/SU-8 sampledescribed above. Image 2103 of FIG. 21 shows the resulting loss tangentimage corrected for squeeze film damping effects. After applying thecorrection shown in image 2103, the loss tangent over the Si is on theorder of 0.01 and over the SU-8 on the order of 0.05. Thus the order isnow correct for the two materials, namely tan δ_(Si)<tan δ_(SU8).

Another improvement in loss tangent imaging is to include energy beingtransferred to higher harmonics of the cantilever. This can be asignificant effect at low Q values. Energy losses to higher harmonics ofthe cantilever are more significant at lower Q than at higher Q. J.Tamayo, et al., High-Q dynamic force microscopy in liquid and itsapplication to living cells, Biophysical J., 81, 526-37 (2001), hasaccounted for this energy dissipation by including the harmonic responseof the cantilever. By extending this analysis to storage, we derived anextension of the SimpleTand expression of the loss tangent that nowincludes harmonic correction terms:

$\begin{matrix}{{\tan \; \delta} = {\frac{{\sin \; \varphi_{1}} - {\sum\limits_{n \geq 1}^{N}{n^{2}\frac{A_{n}^{2}}{A_{1}A_{free}}}}}{{\cos \; \varphi_{1}} - {Q{\sum\limits_{n \geq 1}^{N}{( {n^{2} - 1} )\frac{A_{n}^{2}}{A_{1}A_{free}}}}}}.}} & ({HarmTand})\end{matrix}$

In equation HarmTand, n is the order of the harmonic (ranging from thefundamental at n=1 up to the limit N) and A_(n) the amplitude at the nthharmonic. In the case of the dissipation term (the numerator), theharmonics behave as a “channel” for increased damping. Specifically, ifenergy goes into the harmonics at the fundamental mode, damping willappear to increase. In the case of the storage term (the denominator),energy going into the harmonics looks like a reduction in the kineticenergy of the cantilever. This also has the effect of reducing theapparent storage power in equations FullTand and SimpleTand. The twoeffects act in concert to increase the measured loss tangent.

In addition to measuring many of the harmonics of the loss tangent, theerror associated with energy losses can be estimated and improved uponby simply measuring the response of the cantilever at a harmonic, forexample the 6th or 4th harmonic, that is close to the next highestresonant mode.

Here we discuss two aspects of the behavior of materials that affectloss tangent estimates. The first concerns the viscoelastic response ofa material. Implicit in our definition and use of the loss tangentexpression is the assumption that a sample measured with an AFMindenting tip is a linear viscoelastic material. In this limit, thestrains would remain small and follow a linear stress-strainrelationship, so that an increase in stress would increase strain by thesame factor. However, the assumption of linear viscoelasticity may notalways hold. For instance, the experimental settings may applysufficiently large stresses to cause a nonlinear response. In suchcases, loss tangent measurements will not accurately represent the tan δof the material. One test for the linearity of the response, or improperexperimental parameters in general, is to try to measure the losstangent as a function of indentation depth. In the limit that theinteraction is governed by linear viscoelastic theory, there should notbe a depth dependence in the estimated mechanical parameters such asloss tangent or modulus.

FIG. 29 shows how a loss tangent estimate improves with cantileveroscillation amplitude on a spin-coated film of polypropylene andpolystyrene. In the results described in this figure, the free amplitudeA_(free) was varied while the setpoint ratio α≡A/A_(free) was heldconstant at α=0.5. For a series of A_(free) values two points areplotted against the loss tangent that represents the average of thepolypropylene and polystyrene regions of the corresponding inset images.It can be seen that as A_(free) increases from 70 nm to 290 nm, (points2900, 2910, 2920, 2930 and 2940 in FIG. 29) tan δ decreases andapproaches the expected values of tan δ≈0.1-0.2 for polypropylene andpolystyrene. The inset images 2905, 2915, 2925, 2935 and 2945corresponding to points 2900, 2910, 2920, 2930 and 2940, respectively,display with the same absolute grayscale the values of tan δ, whichrange from tan δ=0 (black) to tan δ=0.5 (white).

The results of FIG. 29 are consistent with the simple idea that as thefree amplitude increases, the tip spends more time interacting withviscoelastic contact forces. Thus, the relative contribution of theseforces to the value of the loss tangent increases, and that valueapproaches the ideal limit. In addition, the measurement uncertainty(indicated in FIG. 29 by error bars of one standard deviation) decreasesdramatically.

Finally, the graph 2950 inserted at the upper right of FIG. 29 shows atypical time evolution of the loss tangent measurement for the largestvalue of A_(free).

The other aspect of the behavior of materials which affects loss tangentestimates that we will discuss concerns the potential for plasticdeformation of the tip of the cantilever or the sample. At the extremelimit of nonlinear viscoelasticity, plastic deformation of the samplerepresents irreversible work that will appear in the numerator of theloss tangent. If the tip does plastic work on the sample, or vice versain the case of hard samples, this will be indistinguishable from sampledissipation (i.e. G″). Thus, plasticity should lead to an overestimationof the loss tangent. The complex theory of indentation-inducedplasticity in materials is strongly dependent on indenter shape and isbeyond the scope of this work. K. L. Johnson, Contact Mechanics(Cambridge University Press, Cambridge, UK, 1985); C. M. Mate, Tribologyon the Small Scale (Oxford University Press, Oxford, UK, 2008) However,we can estimate the yield stress Y from its relation to the forceF_(plastic) needed for plastic deformation:

$\begin{matrix}{{F_{plastic} = {\frac{( {16\; \pi} )^{2}}{6}( \frac{R}{E_{c}} )^{2}Y^{3}}},} & ({FPlastic})\end{matrix}$

where R is the radius of curvature of the hemispherical indenter tip andE_(c) is the elastic modulus.

We can use the equation Fplastic to explore the relationship betweenYoung's modulus and strength for a range of common materials. Here, theterm “strength” corresponds to yield strength for metals and polymers,compressive crushing strength for ceramics, tear strength for elastomersand tensile strength for composites and woods. With R=10 nm andF_(plastic)=1 pN, 1 nN and 1 μN in Fplastic, we find that a significantfraction of materials is expected to plastically yield for loadingforces between 1 and 10 nN, a range typical in AM mode AFM experiments.This illustrates the importance of using the appropriate operatingforces when performing experiments.

Furthermore, plastic deformation can lead to the formation of surfacedebris, which may also result in contamination of the AFM tip. In thiscase, dissipation at the debris-sample and debris-tip interfaces mayprovide additional unwanted contributions to the loss tangentmeasurement. It is therefore important to understand and minimizecontaminants on the surface.

We now consider another method of estimating loss tangent which we referto as the differential loss tangent method.

The impedance of a freely vibrating cantilever

⁻¹(ω) with effective stiffness k_(c), mass m_(c), and damping b_(c)which is driven at some angular frequency ω is

⁻¹(ω)=k _(c) −m _(c)ω² +iωb _(c) (where i=√{square root over (−1)}) witha corresponding resonant frequency ω_(c)=√{square root over (k _(c) /m_(c))} and a quality factor Q_(c)=k _(c)/ω_(c) b _(c). The impedancedescribes the complex-valued driving force F_(d) necessary to excite thecantilever with some oscillation amplitude A. The impedance has units ofN/m. The magnitude spectrum of the cantilever |

(ω)| can be expressed as

${{{(\omega)}} = {\frac{A}{F_{d}} = {{- \frac{\sin \; {\theta_{}(\omega)}}{\omega \; b_{c}}} = {\frac{\omega_{c}}{\omega}\frac{Q_{c}\sin \; \theta_{}}{k_{c}}}}}},$

where θ

(ω) is the phase spectrum given by

${\theta_{}(\omega)} = {{\tan^{- 1}\{ \frac{\omega \; b_{c}}{k_{c} - {m_{c}\omega^{2}}} \}} = {\tan^{- 1}{\{ \frac{\omega \; b_{c}}{k_{c}( {- ( {\omega/\omega_{c}} )^{2}} )} \}.}}}$

It is also convenient to express the magnitude spectrum as:

${{(\omega)}} = {\frac{A}{F_{d}} = {\frac{1}{( {k_{c} - {m_{c}\omega^{2}}} )^{2} + ( {\omega \; b_{c}} )^{2}}.}}$

The impedance

⁻¹(ω) is measured experimentally by exciting the cantilever with adriving force F_(d) through a variety of means known to those skilled inthe art at a range of frequencies and fitting the observed amplitudeA(ω) and phase φ(ω) to the equation

${^{- 1}(\omega)} = {\frac{F_{d}}{A(\omega)}{^{{\varphi}{(\omega)}}.}}$

This fit (or only fitting A(ω)) allows the extraction of the cantileverparameters k_(c), m_(c), and b_(c). Note that

⁻¹(ω)=|

(ω)|⁻¹·

is a theoretical true impedance profile which is inferred by anexperimental measurement

$\frac{F_{d}}{A(\omega)} \cdot {^{{\varphi}{(\omega)}}.}$

In the presence of an actual tip-sample impedance interaction profile,

⁻¹(ω, z), defined as

⁻¹(ω,z)=k _(i)(z)+iωb _(i)(z),

where z is a generalized position coordinate, this profile can beassumed to be the true impedance interaction profile that has not yetbeen subject to convolution due to cantilever amplitude. The impedanceof this interacting cantilever is

_(i) ⁻¹(ω,z _(c))=

⁻¹(ω)+

⁻¹(ω,z)=k _(c) +k _(i)(z _(c))−m _(c)ω² +iω[b _(c) +b _(i)(z _(c))],

where z_(c) is the instantaneous position of the cantilever tip. Panel3610 of FIG. 36 depicts the complex-valued characterization oftip-sample interaction in this situation¹. ¹ Note that Q=1 was used inPanel 3610 of FIG. 36 so that the axes can be scaled 1:1. However, thesame mathematical formulism applies to all Q values.

The impedance interaction can be extracted from a measurement by takingthe difference between an interacting cantilever

_(i) ⁻¹, and the same cantilever at some reference position

_(r) ⁻¹ with no interaction

⁻¹(ω,z)=

_(i) ⁻¹(ω,z _(c))−

_(r) ⁻¹(ω).

Experimentally, this is performed at a single drive frequency ω_(d) atsome distance z_(c) from the sample

${{{k_{i}( z_{c} )} + {{\omega}_{d}{b_{i}( z_{c} )}}} = {{\frac{F_{d}}{A_{i}}^{{\varphi}_{i}}} - {\frac{F_{d}}{A_{r}}^{{\varphi}_{r}}}}},$

where the reference amplitude A_(r) and reference phase φ_(r) aremeasured with no tip-sample interaction at some distance z>>z_(c), andthe interaction amplitude A_(i) and interaction phase φ_(i) are measuredat a distance z_(c). This complex-valued equation can be split into itsreal and imaginary components. Any change in the real components relatesto interaction stiffness

$k_{i} = {{\frac{F_{d}}{A_{i}}\cos \; \varphi_{i}} - {\frac{F_{d}}{A_{r}}\cos \; {\varphi_{r}.}}}$

Changes in the imaginary components relate to interaction damping

${\omega_{d}b_{i}} = {{\frac{F_{d}}{A_{i}}\sin \; \varphi_{i}} - {\frac{F_{d}}{A_{r}}\sin \; {\varphi_{r}.^{2}}}}$

The conserved energy of an interacting cantilever averaged over onecycle is

E _(cons)=½k _(i) A ²+½k _(c) A ²,

while the dissipated energy averaged over one cycle is

E _(diss)=½ωb _(i) A ²+½ωb _(c) A ².

The loss tangent isolates the energy conserved and dissipated by theinteraction alone, resulting in

${\tan \; \delta} = {\frac{E_{{diss},i}}{E_{{cons},i}} = \frac{\omega \; b_{i}}{k_{i}}}$

Assuming some arbitrary, but fixed, drive frequency ω=ω_(d) and somedriving force F_(d)

${{\tan \; \delta} = {\frac{\omega_{d}b_{i}}{k_{i}} = \frac{{\frac{F_{d}}{A_{i}}\sin \; \varphi_{i}} - {\frac{F_{d}}{A_{r}}\sin \; \varphi_{r}}}{{\frac{F_{d}}{A_{i}}\cos \; \varphi_{i}} - {\frac{F_{d}}{A_{r}}\cos \; \varphi_{r}}}}},$

which simplifies to

${\tan \; \delta} = {\frac{{A_{r}\sin \; \varphi_{i}} - {A_{i}\sin \; \varphi_{r}}}{{A_{r}\cos \; \varphi_{i}} - {A_{i}\cos \; \varphi_{r}}}.^{3}}$

-   -   ² Note that taking the difference between        _(i) ⁻¹ and        _(r) ⁻¹ makes the mass term conveniently disappear.    -   ³ Note that the following expressions are mathematically        identical

${\tan \; \delta} = {\frac{{A_{r}\sin \; \varphi_{i}} - {A_{i}\sin \; \varphi_{r}}}{{A_{r}\cos \; \varphi_{i}} - {A_{i}\cos \; \varphi_{r}}} = {\frac{{\alpha\Omega} - {\sin \; \varphi}}{{\alpha \; {Q( {1 - \Omega^{2}} )}} - {\cos \; \varphi}}.}}$

-   -   The advantage of parameterizing the tan δ expression as a        function of (A_(r),A_(i),φ_(r),φ_(i)) as opposed to (Ω,α,Q,φ) is        that (A_(r),φ_(r)) can be measured from a reference image taken        at a soft setpoint, far from the surface or anywhere in between.

Panel 3620 of FIG. 36 depicts how the differential loss tangentmeasurement can probe the local loss tangent of the interaction or theintegrated loss tangent of the interaction, depending on where thereference amplitude and phase are measured.

For convenience we have summarized the steps required to calibrate anAFM for operation in AM mode and measuring loss tangent, followed by theadditional steps required for measuring differential loss tangent. Weassume that the AFM in question is equipped with course-positioning andfine-positioning systems for maneuvering the cantilever and the sample.

Basic AFM Calibration Protocol

-   -   1. Turn drive mechanism (e.g.: photothermal, magnetic or        piezoacoustic excitation) OFF    -   2. Position detection laser on the cantilever tip and preferably        center the photodetector to null the output.    -   3. Calibrate the stiffness of the cantilever from a measured        thermal spectrum far from the surface (at least as far as the        width of the cantilever) and record the natural frequency of the        cantilever.    -   4. Approach the sample and tip to close proximity, as determined        by optical inspection of the tip and sample locations, by using        the coarse-positioning system of the AFM    -   5. Turn drive mechanism ON        -   a. Set the drive frequency equal to the natural frequency of            the cantilever as determined in the thermal (step 3)        -   b. Set drive amplitude to approximate desired free            oscillation amplitude    -   6. Approach surface until contact is established by setting the        feedback set point to a desired interaction amplitude that is        slightly less than the free oscillation amplitude (Alternating        between the fine-positioning system and coarse-positioning        system may yield optimal results)    -   7. Fully retract the sample from the tip of the cantilever        within the range of the fine-positioning system    -   8. Acquire cantilever tune        -   a. Set the drive frequency to the natural frequency of the            cantilever far from the sample, or        -   b. Set the drive frequency to a local amplitude maximum of            the cantilever tune near the cantilever resonant frequency    -   9. Turn drive mechanism OFF and acquire an additional thermal        spectrum of the cantilever that is closer to the imaging        conditions (If photothermal excitation will be used for imaging,        the excitation laser should be turned on during the cantilever        thermal)    -   10. Using a simple harmonic oscillator model and the data from        the thermal spectrum define the new natural frequency of the        cantilever and determine the phase-to-drive-frequency        relationship of the cantilever    -   11. Use the stiffness from the first thermal spectrum acquired        in step 3 to calculate the optical lever sensitivity of the        second thermal spectrum    -   12. Turn drive mechanism ON    -   13. Choose a desired drive frequency        -   a. Optimal drive frequency may be equal to the cantilever            natural or        -   b. Optimal drive frequency may be equal to a local maximum            of the cantilever tune    -   14. Choose drive amplitude        -   a. Set to achieve desired cantilever free oscillation            amplitude        -   b. Record the amplitude as the cantilever reference            amplitude A_(r)    -   15. Set phase from the simple harmonic fit made to the thermal        spectrum in step 10.        -   a. Repeat this step whenever the drive frequency is changed        -   b. Repeat this step whenever the drive amplitude is changed        -   c. If the phase is not within [80°, 100° ], consider            resuming protocol at step 1        -   d. Record phase as the cantilever reference phase φ_(r)            IF the drive frequency is changed, resume at step 1            IF the drive amplitude is changed, resume at step 11            IF the detection laser spot is changed, resume at step 10            IF any parameter of the drive mechanism is changed, resume            at step 9

Data Acquisition Protocol

-   -   1. Approach the sample using the preferred method of distance        feedback protocol        -   a. Closed-loop feedback based on a constant amplitude            setpoint or any other desired setpoint, or        -   b. Open-loop (no feedback) approach of the sample    -   2. Acquire data interaction amplitude A_(i) and data interaction        phase φ_(i) during data acquisition

Data Processing Protocol for Sample Approach Curves

-   -   1. Thermal Reference    -   The foregoing Basic AFM Calibration Protocol provides the        reference parameters (A_(r),φ_(r)) and the Data Acquisition        Protocol provides the data interaction variables (A_(i),φ_(i))        that can be processed by the differential loss tangent equation        to result in a loss tangent measurement tan δ.    -   2. Fixed Reference    -   The reference parameters (A_(r),φ_(r)) may be updated by        selecting a data point (or average of data points) from the data        interaction variables (A_(i),φ_(i)). In that case the        differential loss tangent equation does not require recording        the reference parameters (A_(r),φ_(r)) before data acquisition.        However, the phase does require recording using step 14 so that        the absolute cantilever phase is recorded during the        interaction. Any arbitrary phase offset causes an error in the        tan δ calculation. In contrast, any systematic calibration error        in the amplitude is cancelled out when computing tan δ.    -   3. Moving Reference    -   Alternatively, the reference parameters (A_(r),φ_(r)) may be        updated discretely or continuously as the data interaction        variables (A_(i),φ_(i)) are being processed. That is, every set        of data interaction variables (A_(i),φ_(i)) can be assigned        different reference parameters (A_(r),φ_(r)) when processing the        data using the differential loss tangent equation.

Data Processing Protocol for 2D Images

-   -   1. Thermal Reference    -   A two-dimensional map of interaction variables (A_(i),φ_(i)) can        be processed using the differential loss tangent equation while        using the data reference parameters (A_(r),φ_(r)) recorded        during the calibration protocol.    -   2. Differential Imaging    -   Alternatively, two images can be acquired either consecutively        or simultaneously (by interleaving data acquisition at two        different imaging setpoints) and one image can be used to        determine the interaction variables (A_(i),φ_(i)) and the other        image can be used to determine the data reference parameters        (A_(r),φ_(r)).

Bringing the discussion of loss tangent to a close we now consider howerror and noise affect estimates of loss tangent.

Thermal and other random errors set a minimum detectable threshold forthe loss tangent, typically in the range of tan δ≈10⁻². This randomnoise is typically dominated by Brownian motion of the cantilever forthe Asylum Research Cypher AFM in AM mode. Other random noise such asshot noise might become significant and if it were it would have to beincluded as well.

Systematic errors are typically associated with choosing the appropriatepoint for zero dissipation. It is convenient to divide into two tasks.First, the task of correctly tuning the cantilever at the referenceposition, that is, defining the resonant frequency, the quality factor,the free amplitude and the phase offset to correct for ubiquitousinstrumental phase shifts. In particular, the most accurate estimates ofthe loss

tangent require higher precision in these tune parameters than commonlypracticed in the art. Second, the task of accounting for non-linearviscoelastic contact mechanical forces between the tip and the sample.In particular, we describe improvements in the estimation of the losstangent that comprise (i) a novel method for measuring the cantileverfree amplitude and resonant frequency at every pixel with use of aninterleaved scanning technique and (ii) methods of accounting for thepresence of forces other than linear viscoelastic interactions betweenthe tip and the sample.

To understand how noise estimates affect the loss tangent, it is usefulto map amplitude and phase onto an equation for the loss tangent. FIG.30 shows contour plots of tan δ versus phase φ and relative amplitude aα≡A/A_(free) created with the SimpleTand equation discussed above. Image3020 of FIG. 30 shows the result of this mapping: the loss tangent isconstant along a particular contour, even though the relative amplitudeand phase values continuously vary. The lower boundary (black dottedline) 3015 in image 3020 of FIG. 30 represents the case of zerodissipation, where the interaction between the tip and the sample isperfectly elastic. This curve is given by the zero-dissipationexpression sin φ=α from J. P. Cleveland, et al., Energy dissipation intapping mode atomic force microscopy, Applied Physics Letters, 72, 20(1998). Phase values near this boundary are to be expected whenmeasuring relatively loss-free metals, ceramics and other materials withlow losses and high modulus. Any experimental phase and relativeamplitude values below this boundary represent a violation of energyconservation and presumably indicate improper calibration of thecantilever parameters.

Above the no-loss boundary 3015 in image 3020 of FIG. 30, the solidlines represent contours of constant loss tangent, with values denotedin the legend of the image. At a given relative amplitude α, the phase φmoving towards φ=90° corresponds to an increasing loss tangent. As thevalue of tan δ increases, the contours approach the upper boundary ofφ=90°. This corresponds to all dissipation with no elastic storage, suchas expected for a purely viscous fluid. The other images of FIGS. 30,3010, 3030 and 3040, show SNR contours with various amounts ofsystematic and random noise in the observables, as labeled in Table Ibelow.

It is useful to perform a standard error analysis to understand theimportance of the parameters in the FullTand and SimpleTand and theirimpact on practical loss tangent imaging. As seen from the abovediscussion, there are two types of error to be addressed: random andsystematic.

With standard error analysis, the random uncertainty in loss tangentimaging Δ_(r)(tan δ) due to uncorrelated random uncertainty Δ_(r)φ, isgiven by Δ_(r)Ω, Δ_(r)aα and Δ_(r)Q in φ, Ω, α and Q, respectively, from

${\Delta_{r}( {\tan \; \delta} )} = {\sqrt{\begin{matrix}{{( \frac{{\partial\tan}\; \delta}{\partial\varphi} )^{2}( {\Delta_{r}\varphi} )^{2}} + {( \frac{{\partial\tan}\; \delta}{\partial\Omega} )^{2}( {\Delta_{r}\Omega} )^{2}} +} \\{{( \frac{{\partial\tan}\; \delta}{\partial\alpha} )^{2}( {\Delta_{r}\alpha} )^{2}} + {( \frac{{\partial\tan}\; \delta}{\partial Q} )^{2}( {\Delta_{r}Q} )^{2}}}\end{matrix}}.}$

Similarly, the systematic uncertainty in loss tangent imaging Δ_(s)(tanδ) due to systematic uncertainty Δ_(s)φ, is given by Δ_(s)Ω, Δ_(s)α andΔ_(s)Q respectively, for small uncertainty values from

${\Delta_{s}( {\tan \; \delta} )} = {{\frac{{\partial\tan}\; \delta}{\partial\varphi}\Delta_{s}\varphi} + {\frac{{\partial\tan}\; \delta}{\partial\Omega}\Delta_{s}\Omega} + {\frac{{\partial\tan}\; \delta}{\partial\alpha}\Delta_{s}\alpha} + {\frac{{\partial\tan}\; \delta}{\partial Q}\Delta_{s}Q}}$

For the case of both systematic and random uncertainty, the totaluncertainty is determined by

Δ_(tot)(tan δ)=√{square root over ([Δ_(s)(tan δ)]²+[Δ_(r)(tan δ)]²)}

The separate derivatives in the above two equations for random andsystematic uncertainty can be evaluated as

$\frac{{\partial\tan}\; \delta}{\partial\varphi} = \frac{1 + {\alpha \; {Q( {\Omega^{2} - 1} )}\cos \; \varphi} - {{\alpha\Omega}\; \sin \; \varphi}}{\lbrack {{\alpha \; {Q( {\Omega^{2} - 1} )}} + {\cos \; \varphi}} \rbrack^{2}}$${\frac{{\partial\tan}\; \delta}{\partial\alpha} = \frac{{\Omega \; \cos \; \varphi} + {{Q( {\Omega^{2} - 1} )}\sin \; \varphi}}{\lbrack {{\alpha \; {Q( {\Omega^{2} - 1} )}} + {\cos \; \varphi}} \rbrack^{2}}},{\frac{{\partial\tan}\; \delta}{\partial\Omega} = \frac{{\alpha \; {Q\lbrack {{\alpha ( {\Omega^{2} + 1} )} - {2{\Omega sin}\; \varphi}} \rbrack}} - {\alpha \; \cos \; \varphi}}{\lbrack {{\alpha \; {Q( {\Omega^{2} - 1} )}} + {\cos \; \varphi}} \rbrack^{2}}}$and$\frac{{\partial\tan}\; \delta}{\partial Q} = {\frac{\alpha \; ( {\Omega^{2} - 1} )( {{\alpha\Omega} - {\sin \; \varphi}} )}{\lbrack {{\alpha \; {Q( {\Omega^{2} - 1} )}} + {\cos \; \varphi}} \rbrack^{2}}.}$

As a specific example comparing the relative weights of the foregoingterms, we assume a cantilever similar to those used in the experimentsdiscussed above, the AC240 from Olympus, with the following operatingparameters: Ω=1, α=0.5, Q=150 and ω=2π75 kHz. These conditionscorrespond to operating on resonance with a setpoint ratio of 0.5,(i.e., AM mode amplitude is 50% of the free amplitude). The sample losstangent is assumed to be tan δ=0.1, as expected for a polymer such ashigh-density polyethylene. From the FullTand equation, these valuesyield a cantilever phase shift of φ≈36° (refer to image 3010 of FIG. 30to see this graphically). With the four equations immediately above, theindividual error terms can be evaluated as

${\frac{{\partial\tan}\; \delta}{\partial\varphi} = {0.019\text{/}{^\circ}}},{\frac{{\partial\tan}\; \delta}{\partial\alpha} = {- 1.24}},{\frac{{\partial\tan}\; \delta}{\partial\Omega} = {{{- 20.7}\mspace{14mu} {and}\mspace{14mu} \frac{{\partial\tan}\; \delta}{\partial Q}} = 0.}}$

In the experience of the inventors, typical values of random uncertaintyin amplitude, phase and tuning frequency are Δ_(s)φ=0.3°, Δ_(s)α=10⁻²,and Δ_(s)Ω=10⁻³, respectively, for the first mode. These values yield atotal loss tangent error estimate of Δ_(s) tan δ=0.026. This result issurprisingly large, given that the loss tangent of many polymermaterials is of similar order or smaller. Of the three contributions,the largest is from the uncertainty in tuning frequency Δ_(s)Ω. If thatis improved to Δ_(s)Ω=10⁻⁴, the loss tangent uncertainty is reduced toΔ_(s) tan δ=0.016. This somewhat unexpected result implies that carefulidentification of the cantilever resonance frequency is important forquantitative loss tangent estimation. While this level of tuninguncertainty is not typical for tapping mode operation, it is reasonable.An informal survey of autotune functions on a variety of commercial AFMinstruments yielded scatter as large as 1 kHz and typically on the orderof several hundred hertz. In the course of these investigations, we haveimproved our tuning routine to bring the frequency uncertainty closer toa few tens of hertz, Δ_(s)Ω≦10⁻⁴, without adding significant time to theprocedure.

It is also useful to define the signal to noise ratio (SNR) to considerhow SNR affects the loss tangent:

$\begin{matrix}{{SNR} = {\frac{\tan \; \delta}{\Delta ( {\tan \; \delta} )}.}} & ({SNRDef})\end{matrix}$

Typically, we would like SNR≧1 so that the signal is larger than theuncertainty. For the example above with the AC240 Olympus cantilever,this corresponds to tuning the cantilever to within ˜300-400 Hz of the75 kHz resonance. To better visualize this, images 3020, 3030 and 3040of FIG. 30 show curves of constant SNR as a function of relativeamplitude α and phase φ. The SNR curves were determined by finding thevalues of α and φ that simultaneously satisfy the SimpleTand and SNRequations for a given value of SNR. The values in Table I below fordifferent elements of measurement uncertainty were used in the fourequations immediately above the SNR equation to obtain the overalluncertainty Δ_(tot)(tan δ). For this purpose it is assumed that therandom phase error is dominated by thermal motion with Δ_(r)φ=0.5°. Theoverall uncertainty increases from the thermal-noise-limited case inimage 3020 of FIG. 30 to a case with both random and systematic errorsin image 3040 of FIG. 30. As the uncertainty increases, the accessibleregion of (α,φ) space for a given SNR decreases significantly. Thehighest errors occur at the boundaries of the physically meaningfulphase and amplitude values. Near the no-loss boundary, the uncertaintycontributions are larger than the loss tangent value itself, asdiscussed above. For large loss tangents, note that the tangent functiondiverges at 90°. Thus small errors in phase or amplitude can lead toenormous errors in the estimated loss tangent.

TABLE I Image Number Δ_(r)Ω Δ_(s)Ω Δ_(r)φ[°] Δ_(s)φ[°] 3010 0 0 0 0 3020and 3050 0 0 0.5 0 (3025) 3030 and 3050 10⁻⁵ 10⁻⁴ 0.5 0 (3035) 3040 and3050 10⁻⁵ 10⁻⁴ 0.5 2 (3045)Error parameters for FIG. 30. The values used in each image of FIG. 30for the random (subscript r) and systematic (subscript s) errors inrelative frequency ΔΩ and phase Δφ. Other error sources are heldconstant as follows: Δα_(r)=Δα_(s)=0 and ΔQ_(r)=ΔQ_(s)=0.

The graph 3050 in FIG. 30 also shows the SNR curves from the errorconditions listed in Table 1 and illustrated in images 3020, 3030 and3040 of FIG. 30 made at a setpoint of α=0.5. As noted in Table 1, curve3025 shows the minimum noise limited by fundamental thermal physics ofthe cantilever while curves 3035 and 3045 show the effects of increasingrandom and systematic noise. It is clear from curve 3025 that even inthe most optimistic experimental case, thermal noise still limits losstangent measurements on materials to tan δ{tilde under (<)}10⁻².

We now turn to a further discussion of Frequency Modulation (FM) AFM.With FM AFM we measure the frequency shift as the tip interacts with thesurface, and we can therefore quantify tip-sample interactions.

the frequency shift of a cantilever in FM mode is given by

$\begin{matrix}{{\Delta \; f_{2}} = {f_{0,2}\frac{\langle{F_{ts}z}\rangle}{k_{2}A_{2}^{2}}}} & ({FreqShift})\end{matrix}$

In this equation Δf₂ is the shift of the second resonant mode as the tipinteracts with the surface, f_(0,2) is the second resonant frequencymeasured at a “free” or reference position, k₂ is the stiffness of thesecond mode and A₂ is the oscillation amplitude of the second mode as itinteracts with the surface. F_(ts) and z have previously been defined inconnection with the presentation of the FullTand equation above. TheFreqShift equation, as with the expression for the loss tangent, doesnot directly involve optical lever sensitivity.

In the limit that the oscillation amplitude of the second mode is muchsmaller than the oscillation amplitude of the first mode and is alsomuch smaller than the length scale of F_(ts), the tip-sample interactionforce, we can relate the measured frequency shift Δf₂ to tip-samplestiffness:

$\begin{matrix}{{\Delta \; f_{2}} \approx {\frac{f_{0,2}}{2}{\frac{k_{ts}}{k_{2\;}}.}}} & (5)\end{matrix}$

This can be solved for the tip-sample stiffness as

$k_{ts} \approx {\frac{2\Delta \; f_{2}}{f_{{0,2}\;}}.}$

The second mode resonant behavior provides a direct measure of thetip-sample interaction forces as shown in FIG. 23. In the graph on theleft side of FIG. 23 the amplitude 2301 of the first mode resonancemeasured on the y-axis is plotted versus tip-sample distance measured onthe x-axis. As the cantilever approaches the surface, it firstexperiences net attractive forces at point 2302 which reduce theamplitude. Thereafter, there is a switch at point 2303 from netattractive to net repulsive interactions which results in adiscontinuity in the amplitude curve. This effect is well-known in theliterature. Beyond point 2303, the interaction is dominated by repulsiveforces and the curve 2304 is somewhat different.

In a second graph on the left side of FIG. 23 the phase 2305 of thefirst mode resonance, which is measured simultaneously with theamplitude of the first mode resonance versus tip-sample distance curve,gives additional information. During the same interval when theamplitude versus tip-sample distance curve experiences net attractiveforces which reduce its amplitude, the phase curve also experiencesforces which ultimately reduce its amplitude 2306. Similarly when theamplitude versus tip-sample distance curve experiences a switch from netattractive to net repulsive interactions resulting in a discontinuity inthe amplitude curve the phase curve experiences a similar discontinuity2307. Beyond point 2307, the interaction is dominated by repulsiveforces and the curve 2308 is somewhat different, just as with theamplitude curve. Note that both transitions 2303 with the amplitudeversus tip-sample distance curve and 2307 with the phase curve have somehysteresis between the approach and retract portions of the curves.

In the graph on the right side of FIG. 23 the families of the amplitudecurves 2311 and 2312 of the second mode resonance measured on the y-axisare plotted versus the drive frequency of the cantilever measured on thex-axis. In a second graph of FIG. 23 located above the families ofamplitude graph, the families of phase curves 2309 and 2310 of thesecond mode resonance measured on the y-axis versus the drive frequencyof the cantilever measured on the x-axis. Both the amplitude and phasefamilies of second mode resonance were measured while the first mode washeld essentially constant by adjusting the z-height of the cantileverwhile the second mode frequency was ramped using the 1090 drive of theapparatus for probing flexural resonances depicted in FIG. 20.Alternatively the second mode frequency tunes could be made by chirping,band-exciting, intermodulating or exciting in some other manner thatexplores the frequency content of the interaction. At the same time,amplitude 2315 and phase 2314 tunes far from the surface can be taken asshown in the graph on the right side of FIG. 23. During the attractiveportion of the interaction, the phase curve 2309 will move to the left,indicating that the resonance has shifted to a lower frequency, asexpected from an attractive interaction. During the repulsive portion ofthe interaction, the phase curve 2310 will move to the right, a shift toa higher frequency as expected from a stiffer interaction. The amplitudecurves show corresponding behavior, with the peak of curve 2311 shiftingto the left toward lower frequencies) during the attractive portion ofthe interaction and the peak of curve 2312 shifting to the right towardhigher frequencies during the repulsive portion of the interaction. Inaddition, the peak values of the amplitude curves 2311 and 2312decrease, indicating an overall increase in tip-sample damping ordissipation.

An alternative method for extracting the second mode resonancemeasurements just presented is to use the phase locked loop (PLL) 1160of the apparatus for probing flexural resonances depicted in FIG. 20.Using this alternative the amplitude and phase families of second moderesonance would be measured while holding the first mode essentiallyconstant by adjusting the z-height of the cantilever while the secondmode frequency was ramped using the 1090 drive. The frequency of thesecond mode can be directly measured by engaging a phase locked loop(PLL) 1160 and repeating the amplitude versus distance experiment, whilesimultaneously recording the second mode resonant frequency output ofthe PLL. The result of the output from f₂ 2010 and A₂ 2013 of the FIG.20 apparatus is the curve 2313 located between the families of amplitudecurves 2311 and 2312 of the second mode resonance and the families ofphase curves 2309 and 2310 of the second mode resonance. This resultimplies that the FIG. 20 apparatus with its PLL can be used, at least inthis case, as a rapid means of interrogating the cantilever as to itssecond mode response. Note that other methods for extracting the secondmode resonance measurements could also be used, including the DARTmethod discussed above, the Band Excitation method developed at the OakRidge National Laboratory, chirping, intermodulation and othertechniques that provide information about frequency response.

One situation where it is advantageous to omit the use of the PLL iswhen the cantilever actuation mechanism includes a frequency-dependentamplitude and phase transfer function. This prerequisite may exist witha large variety of cantilever actuation means including acoustic,ultrasonic, magnetic, electric, photothermal, photo-pressure and othermeans known in the art. Operation in this mode is essentially bimodal,and is variously called DualAC mode, or AM/AM. The principal inventionsdescribing this mode are U.S. Pat. No. 7,921,466, Method of Using anAtomic Force Microscope and Microscope; and U.S. Pat. No. 7,603,891,Multiple Frequency Atomic Force Microscope.

As noted above the phase shift which occurs when the amplitude and phaseof the second mode resonance were measured while the first mode was heldessentially constant by adjusting the z-height of the cantilever whilethe second mode frequency was ramped is accompanied by a stifferinteraction during the repulsive portion of the interaction. We canexploit this relationship between the phase shift and stiffness of thesample starting with the relationship between the frequency and phaseshifts for a simple harmonic oscillator:

${\frac{\partial\varphi}{\partial\Omega} = \frac{( {\Omega^{2} + 1} )Q}{{( {\Omega^{2} - 1} )^{2}Q^{2}} + \Omega^{2}}},$

where Ω≡f_(drive)/f₀ is the ratio of the drive frequency to the resonantfrequency. Using the SimpleTand equation, this relationship can bemanipulated to give the tip-sample interaction stiffness in terms of thephase shift measured at a fixed drive frequency:

$k_{ts} = {\approx {\frac{1}{\Omega}\frac{\Delta\varphi}{Q}{k_{2}.}}}$

It will be noted that this expression is only valid for small frequencyshifts.

The expression of tip-sample interaction stiffness in terms of the phaseshift can be extremely sensitive, down to the level of repeating imagesof single atomic defects. In FIG. 26, two repeating images 2601 and 2602of the surface of calcite in fluid are shown at the left, each with thesame defect, 2603 and 2604, respectively, visible. The correspondingstiffness images, 2605 and 2606, are shown on the right, each with thesingle atomic stiffness defect, 2607 and 2608, respectively, visible inimage. The graph under the images shows line sections 2609 taken acrossthe images with the sub-nanometer peak in the stiffness imagesassociated with the defect clearly visible. This very high resolutionmechanical property measurement in fluid represents a very significantadvance in materials properties measurements using AFM.

Since loss tangent is measurable using the first resonant mode of thecantilever and the frequency shift just described is measured using thesecond resonant mode, both measurements can be made simultaneously. Wehave found however that there are some practical experimental conditionsto consider when applying this bimodal technique to nano-mechanicalmaterial property measurements. To begin with, the cantilever tip issensitive to the G′ and G″ factors of the FullTand equation only inrepulsive mode. This means that the following conditions favoringrepulsive mode should be present:

-   -   1. larger cantilever amplitudes (>1 nm),    -   2. stiffer cantilevers (>1 N/m),    -   3. sharp cantilever tips, and    -   4. lower setpoints.

Furthermore, in net-repulsive mode, the phase of the first mode shouldalways be <90° and typically <50° for most materials; good feedbacktracking (that is, for example, avoiding parachuting and making suretrace and retrace match) is important to assure good sampling ofmechanical properties; and careful tuning of the cantilever resonancesis particularly important to assure the accuracy of both techniques.Relative to this last point, the error for the resonances should be <10Hz and the phase should be within 0.5 degrees. These are more stringentconditions than usual for AM mode but are well within the capabilitiesof commercial AFMs, given proper operation.

FIG. 25 shows an example of a simultaneous loss tangent measurement ofan elastomer-epoxy sandwich in Panel 2501 and a stiffness measurement ofthis sandwich in Panel 2502. The sandwich was assembled by bonding anatural rubber sheet to a latex rubber sheet with an epoxy. The sandwichwas then micro-cryotomed and imaged. From macroscopic measurements, theelasticities were estimated to be ˜40 MPa for the natural rubbercomponent; 4 GPa for the epoxy; and 43 MPa for the latex rubber, all asmeasured with a Shore durometer. Again with macroscopic measurements,loss tangents were estimated to be 1.5, 0.1:2 and 2, respectively,measured with a simple drop test.

The measurements shown in FIG. 25 were made with an Olympus AC160cantilever with a fundamental resonance of 310 kHz and a second moderesonance of 1.75 MHz. The histograms of the loss tangent measurement inPanel 2503 and the stiffness measurement in Panel 2504 show clearseparations of the three components. It should be noted that the surfaceroughness of the sandwich was on the order of 500 nm and despite thisthe materials are clearly differentiated.

The stiffness measurement made with the second resonant mode depends oninteraction stiffness k_(ts). The modulus of the sample(s) in questioncan be mapped by applying a mechanical model. One of the simplest modelsis a Hertz indenter in the shape of a punch. In this case, theelasticity of the sample is related to the tip-sample stiffness by therelation k_(ts)=2aE′, where a is a constant contact area. Combining thiswith the SimpleTand equation above results in the expression

$E^{\prime} = {\frac{\Delta \; f_{2}k_{2}}{{af}_{0,2}}.}$

Thus if the contact radius and spring constant are known, the samplemodulus can be calculated.

Other tip shapes could be used in the model. Calibration of the tipshape is a well-known problem. However, it is possible to use acalibration sample that circumvents this process. As a first step, wehave used a NIST-traceable ultra high molecular weight high densitypolyethelene (UHMWPE) sample to first calibrate the response of theOlympus AC160 cantilever.

The equation immediately above can then be rewritten as E′=C₂Δf₂, whereC₂ is a constant, measured over the UHMWPE reference that relates thefrequency shift to the elastic modulus. The result can be applied tounknown samples.

The foregoing bimodal technique providing for measuring stiffness withthe second resonant mode can be performed at high speeds with the use ofsmall cantilevers. The response bandwidth of the i_(th) resonant mode ofa cantilever is BW_(i)=πf_(i,0)/Q_(i) where f_(i,0) is the resonantfrequency of the i_(th) mode and Q_(i) is the quality factor. Theresonant frequency can be increased without changing the spring constantby making smaller cantilevers. In contrast to normal AM imaging,however, the second resonant mode must still be accessible to thephotodetector—which requires f_(2,0)<10 MHz for an AFM like the AsylumResearch Cypher AFM.

FIG. 24 shows the effect of using smaller cantilevers. Panel 2401 andPanel 2402 show measurements at a 2 Hz line scan rate and a 20 Hz linescan rate, respectively, of a EPDH/Epoxy cryo-microtomed boundary. Thesemeasurements (5 μm on a side) were acquired with an AC55 cantilever fromOlympus (f_(1,0)≈1.3 MHz, f₁₂₀≈5.3 MHz). Panel c of FIG. 24 showshistograms of the Panel 2401 2 Hz measurement (histogram 2403) and thePanel 2402 20 Hz measurement (histogram 2404). The very small deviationbetween the peak elasticity measurements indicates that the high speedimage acquisition did not significantly affect the results.

In general, one can choose any higher resonant mode for measuringstiffness, but the following considerations should be kept in mind:

1. Avoid modes that are at or very close to integer multiples of thefirst resonance. Integer multiples result in harmonic mixing between themodes which can cause instabilities. For example, it is quite commonthat the second resonance of an Olympic AC240 is ˜6× the first. For thatreason, it is desirable to use the third resonant mode instead, whichtypically has a resonance of ˜15.5× the first mode. This point should bekept in mind even when the cantilever is not being driven at the secondresonant mode. If a higher mode is too close to an integer multiple ofthe drive frequency, unwanted harmonic coupling can take place thatleads to spurious, noisy or difficult to interpret results.

2. The sensitivity is optimized when the stiffness of the mode is tunedto the tip-sample stiffness. FIG. 27 shows the effects of choosing theresonant mode that was softer, matched or stiffer than the tip-samplestiffness. An Olympus AC200 cantilever was used to make each of themeasurements shown in FIG. 27. The fundamental resonance of thiscantilever (unlike the other resonant modes, not shown) was at ˜1.15kHz. The second resonance 2701 was at ˜500 kHz, the third 2702 was at˜1.4 MHz, the fourth 2703 at ˜2.7 MHz, the fifth 2704 at ˜4.3 MHz andthe sixth 2705 was at ˜6.4 MHz. The elasticity images are shown inPanels 2706, 2707, 2708, 2709 and 2710 in ascending order by resonance.Note that the contrast in the second mode (soft) Panel 2706 isrelatively low, as is the contrast in the 5th and 6th modes (stiff)Panels 2709 and 2710. This is explicitly visible in the Elasticityhistograms shown in Panels 2711, 2712, 2713, 2714 and 2715, againarranged in ascending order by resonance.

In order to optimize the mechanical stiffness, contrast and accuracy ofthe tip-sample it appears to be advantageous to tune the amplitude ofthe second mode so that it is large enough to be above the detectionnoise floor of the AFM, but small enough not to affect significantly thetrajectory and behavior of the fundamental mode motion as discussedabove in reference to FIG. 4. For this purpose, it is useful to plot thehigher mode amplitude or phase or frequency, or all of them, as afunction of the second mode drive amplitude. This can be done forexample, while the first mode is used in a feedback loop controlling thetip-sample separation or in a pre-determined tip-sample position. Asshown in FIG. 4, careful observation of first mode amplitude, 4010 and4050, and phase, 4020 and 4060, together with the higher mode amplitude4030 and phase 4040 as a function of the higher mode drive amplitude1090 (from FIG. 20), the user can chose a drive amplitude that optimizesthe signal to noise while minimizing the effect of nonlinearities on themeasured signals.

It is also desirable to quantify the measurement of higher modestiffness. In general, this is a challenging measurement. One method isto extend the thermal noise measurement method to higher modes. Thethermal measurement depends on accurately measuring the optical leversensitivity. This can be done by driving each resonant mode separatelyand then measuring the slope of the amplitude-distance curve as thecantilever approaches a hard surface. This calibrated opticalsensitivity can then be used in a thermal fit as is well known in theart to get the spring constant for that particular resonant mode.Typical fits for the fundamental 2801, second resonant mode 2802 andthird resonant mode 2803 are shown in FIG. 28. The optical leversensitivities yielded by these fits were then used to fit the thermalnoise spectra to extract the spring constants for the first 2804, second2805 and third 2806 modes of the Olympus AC240 cantilever.

To obtain the interaction force between the tip and the sample, theFreqShift equation above has to be inverted. Initially, this would bedone where the cantilever amplitude was either much smaller or muchlarger than the length scale of the tip-sample interaction. These limitscan lead to practical errors in experimental data since the length scaleof the tip-sample interaction forces is not necessarily known a priori.However, a very accurate inversion method has been developed by Saderand Jarvis which uses fractional calculus. Recently, Garcia and Heruzohave extended this method to bimodal frequency shifts. In particular,their method connects the frequency shift of the first and secondresonant modes to a force model.

$\begin{matrix}{{\Delta \; f_{1}} = {\frac{f_{1}}{2k_{1}\sqrt{2\pi \; A_{1}}}D^{1/2}{\_ F}( d_{\min} )}} & {2(a)} \\{{\Delta \; f_{2}} = {{- \frac{f_{2}}{2k_{2}\sqrt{2\pi \; A_{1}}}}I^{1/2}{\_ F}( d_{\min} )}} & {2(b)}\end{matrix}$

In these equations and D½ and I½ represent the half-derivative andhalf-integral operators respectively. F(d_(min)) is the tip-interactionforce as a function of the position of closest d_(min) for the tip tothe sample. These expressions are valid in the limit that the highermode amplitude is much smaller than the first and also larger than thecharacteristic length scale of the tip-sample force.

These equations involve frequency shifts in the resonant modes of thecantilever. As such, they are directly applicable to the FM-FM imagingdescribed by Garcia and Herruzo. For other bimodal approaches discussedabove (specifically AM/AM and AM/FM) another step is required since afrequency shift observable has been replaced by a phase observable.Earlier we presented a linear approximation. Now we develop an exactconversion between the phase observable and the frequency shiftobservable of a simple harmonic oscillator (SHO).

For a SHO, the phase and frequency are related by tanφ=f₀f_(i)/Q_(tot)(f₀ ²−f_(i) ²). In this expression, we have designatedthe quality factor as Q_(tot) since it includes both the intrinsicdamping of the cantilever and tip-sample dissipation. This expressioncan be inverted to yield frequency as a function of phase

$\begin{matrix}{\frac{\Delta \; f}{f_{0}} = {{\frac{{\pm 1} + \sqrt{1 + {4Q^{2}\tan^{2}\varphi_{i}}}}{2Q_{i}\tan \; \varphi_{i}} - 1} = {{\pm \frac{1}{2Q_{i}\tan \; \varphi_{i}}} + {{{Order}( \frac{1}{( {Q_{i}\tan \; \varphi_{i}} )^{2}} )}.}}}} & ({FreqCalc})\end{matrix}$

For many samples, tip-sample dissipation is quite small, so that it canbe neglected and the phase shift attributed only to conservativetip-sample interactions. In this situation, the Q factor for the aboveexpression can be estimated from the free-air expression,Q_(tot)≈Q_(c,i), where Q_(c,i) is the quality factor of the i_(th) modeof the cantilever, measured at a reference position close to the sample.It can be shown that this expression is valid when the loss tangent ofthe tip-sample interaction is small, of the order of the cantileverphase noise, tan δ_(i)≦10⁻². For larger tip-sample losses, thetip-sample dissipation must be included in the calculation. This reducesthe quality factor by

Q _(tot,i) =Q _(c,i)/(Q _(c,i) tan δ+1).  (QCalc1)

Using the loss tangent from above, if the cantilever is being driven atits free resonance frequency the tapping mode amplitude and phaseobservable yields:

$\begin{matrix}{Q_{{tot},i} = {\frac{Q_{c,i}\cos \; \varphi_{i}}{{\cos \; \varphi_{i}} + {Q_{c,i}( {{A_{i}/A_{{free},i}} - {\sin \; \varphi_{i}}} )}}.}} & ( {{QCalc}\mspace{14mu} 2} )\end{matrix}$

As noted, this expression is valid when operating at a free resonancefrequency, but is trivially generalizable to the off-resonance case.Using this formalism, it is possible to estimate the equivalentfrequency shift from phase and amplitude measurements. We demonstratethis conversion for the second resonant mode of a cantilever interactingwith a two-component polymer film in FIG. 31.

In FIG. 31 second mode observables were measured as a function of thefirst mode amplitude over a polymer sample composed of polystyrene 3110and 3120 and polypropylene 3130 and 3140. One curve of each polymer 3110and 3140 was made with the phase locked loop engaged. These curvesrepresent a direct measurement of the second mode frequency shift. Themethod described above is verified by comparing the frequency shiftscalculated from the measured second mode phase shifts combined withequations FreqCalc and QCalc1 or QCalc2. Curve 3130 shows the directlymeasured frequency shift while curve 3140 shows the calculated frequencyshift in each case over the lower modulus polypropylene. Similarly,curve 3110 shows the directly measured frequency shift while curve 3120shows the calculated frequency shift over the higher moduluspolystyrene. As these curves make clear, the calculated values agreevery well with the directly measured frequency shifts, verifying andjustifying this approach to estimating equivalent frequency shifts fromphase data.

FIG. 32 summarizes a general approach to obtaining quantitativemechanical properties in AM (or tapping) mode. For illustration, we usethe example of an elastic Hertzian contact model with two unknownparameters: the effective modulus and the indentation depth. The processis quite similar for other models including those with dissipativecontributions as will be apparent to one skilled in the art.

There are many contact mechanics models that may be used forinterpreting tip-sample interactions between a nano-scale indenter tipand a sample. Equations 2(a) and 2(b) provide a framework for applyingthese models to any tip-sample interaction that is fractionallydifferentiable.

An example of a power-law force between the tip-sample indentation andthe applied force was postulated by Oliver and Pharr. Values of m=1, 3/2and 2 are associated with the Hertzian Punch, Sphere and Cone (Sneddon)models respectively. In addition, the α_(m) prefactor is replaced by theappropriate values in the expressions below:

$\begin{matrix}{{{F_{ts}(\delta)} = {E_{m}^{eff}\alpha_{m}\delta^{m}}},} & (3) \\{{{F_{punch}(\delta)} = {2E_{punch}^{eff}R\; \delta}},} & ( {4a} ) \\{{F_{sphere}(\delta)} = {\frac{4}{3}E_{sphere}^{eff}\sqrt{R}\delta^{\frac{3}{2}}}} & ( {4b} ) \\{{F_{cone}(\delta)} = {\frac{2}{\pi}\frac{E_{cone}^{eff}}{\tan \; \theta}\delta^{2}}} & ( {4c} )\end{matrix}$

Expression 3 just above can be applied to the fractional derivatives andintegrals expressions 2(a) and (2b) above. Evaluation of the resultgives the following model-dependent expressions for the indentationdepth:

$\begin{matrix}{\delta_{m} = {\frac{A_{1}}{2}\frac{k_{1}}{k_{2}}\frac{f_{2}}{\Delta \; f_{2}}\frac{\Delta \; f_{1}}{f_{1}}\frac{G_{3}}{G_{1}}}} & {4(a)} \\{{\delta_{cone} = {\frac{3A_{1}}{4}\frac{k_{1}}{k_{2}}\frac{f_{2}}{\Delta \; f_{2}}\frac{\Delta \; f_{1}}{f_{1}}}},} & {4(b)} \\{{\delta_{sphere} = {A_{1}\frac{k_{1}}{k_{2}}\frac{f_{2}}{\Delta \; f_{2}}\frac{\Delta \; f_{1}}{f_{1}}}},} & {4(c)} \\{{\delta_{punch} = {\frac{3A_{1}}{4}\frac{k_{1}}{k_{2}}\frac{f_{2}}{\Delta \; f_{2}}\frac{\Delta \; f_{1}}{f_{1}}}},} & {4(d)}\end{matrix}$

Here, the constants G₁ and G₃ from equation 4 (a) are defined as

$G_{1} = \frac{\Gamma ( {m + 1} )}{\Gamma ( {m + \frac{1}{2}} )}$and$G_{3} = {\frac{\Gamma ( {m + 1} )}{\Gamma ( {m + \frac{3}{2}} )}.}$

Similarly, the effective sample modulus is given by:

$\begin{matrix}{E_{m}^{eff} = {\frac{\pi \; f_{1}^{m - {1/2}}k_{2}^{m + {1/2}}}{\alpha_{m}A_{1}^{m - 1}f_{2}^{m + {1/2}}k_{1}^{m - {1/2}}}\frac{\Delta \; f_{2}^{m + {1/2}}}{\Delta \; f_{1}^{m - {1/2}}}\frac{G_{3}^{m - {1/2}}}{G_{1}^{m + {1/2}}}}} & \; \\{{E_{cone}^{eff} = {\frac{3\pi \sqrt{2}\tan \; \theta \; f_{1}^{3/2}k_{2}^{5/2}}{5\sqrt{5}f_{2}^{5/2}k_{1}^{3/2}}\frac{\Delta \; f_{2}^{5/2}}{\Delta \; f_{1}^{3/2}}}},} & {5(a)} \\{{E_{sphere}^{f} = {\frac{\sqrt{8}f_{1}k_{2}^{2}}{\sqrt{{RA}_{1}}f_{2}^{2}k_{1}}\frac{\Delta \; f_{2}^{2}}{\Delta \; f_{1}}}},} & {5(b)} \\{E_{punch}^{eff} = {\frac{\pi \sqrt{2}f_{1}^{1/2}k_{2}^{3/2}}{R\sqrt{3}f_{2}^{3/2}k_{1}^{1/2}}\frac{\Delta \; f_{2}^{3/2}}{\Delta \; f_{1}^{1/2}}}} & {5(c)}\end{matrix}$

Note that Equations 4 (b) and 5 (b) for the sphere model are identicalto the expressions appearing in the Garcia and Tomas-Herruzo reference.

The equations above involve frequency shifts in the resonant modes ofthe cantilever. In bimodal and AM/FM measurements, the phase measurementof both or the first modes are measured (rather than controlled by aphase locked loop) while the drive frequency is kept constant.

One desirable quality of modulus measurements made in tapping mode isthat the measured modulus be independent of the A₁ setpoint value.

To test this condition the results of a series of numerical simulationsof AM/AM (bimodal) tapping mode amplitude versus distance curves areshown in FIG. 33. The simulations were made with a publicly availabletool: VEDA from nanohub.org. Similar simulations can be made using othersoftware platforms such as MATLAB (Mathworks) and Igor (Wavemetrics).The simulated cantilever had the following parameters: k₁=40 N/m,k₂=1500 N/m, f1=320 kHz, f2=1.8 MHz, R=50 nm, and Afree=10 nm. Thetip-sample interaction was modeled as a spherical Hertzian contactF_(sphere)=4/3E_(sphere) ^(eff)√{square root over (R)}δ3/2. The samplemodulus E_(sphere) ^(eff) was set at 100 MPa trace 3210, 1 GPa trace3220, 10 GPa trace 3230 and 100 GPa trace 3240.

Since the simulation was made in AM/AM mode, the amplitude and phaseobservables of both resonant modes were converted to effective resonantfrequency shifts as discussed above.

This return of a substantially constant modulus versus a indicates thatthe tip-sample contact mechanical model is behaving properly. In theexample here, the implication is that we have correctly chosen thesphere model, consistent with the model used to generate the numericaldata. Tip shape characterization is a well known problem in the art andbased on this observation we describe a new method of characterizing thetip shape below.

FIG. 33 shows a series of curves where the modulus calculated with theabove equations is plotted versus the first mode amplitude, 160 nm. Foreach of the curves, we chose the value at this amplitude to represent a“reference” point of 2 GPa. As discussed above, it is desirable that themodulus is as close to this value for every value of the first modeamplitude. The three standard models—a punch, a sphere and a cone showdiffering success at attaining this goal. The cone curve 3430 deviatesthe most and the sphere curve 3320 is only somewhat more successful. Thepunch curve 3410 in contrast is close to constant over the range offirst mode amplitudes. However, the punch result can be improved upon byallowing the exponent value m to vary. In this case, by choosing m=1.1,we get curve 3440, which is an optimized value of the modulus over thefull range of the measured first mode amplitude. The implication is thatthe tip shape is somewhere between a punch and a sphere, a slightlyrounded punch.

The estimation of loss tangent can be extended to sub-resonantoscillatory measurements as well. While sub-resonant measurementsgenerally have reduced signal to noise performance and tend to exertlarger forces on the sample, they can also have some advantages overresonant techniques.

With sub-resonant oscillatory measurements, we can define the virial ofthe tip-sample interaction by

$V_{ts} = {\frac{\omega}{2\pi}{{\langle{F_{ts}z_{tip}}\rangle}.}}$

Similarly, the dissipated power is given by P_(ts)=<F_(ts)ż_(tip)>.Using these expressions, the loss tangent can be estimated by

${\tan \; \delta} = {\frac{P_{ts}}{V_{ts}}.}$

These equations can be applied to non-resonant oscillatory motionincluding the sub-resonant oscillatory motion made while performingforce-distance curves. These curves include so called “fast” forcecurves that are made at a frequencies greater than a few Hertz up tofrequencies at or near the resonance frequency of the cantilever. Oneadvantage of this approach is that the oscillatory waveform can besinusoidal, triangular or other waveforms that could be expressed as aFourier series.

Although only a few embodiments have been disclosed in detail above,other embodiments are possible and the inventors intend these to beencompassed within this specification. The specification describesspecific examples to accomplish a more general goal that may beaccomplished in another way. This disclosure is intended to beexemplary, and the claims are intended to cover any modification oralternative which might be predictable to a person having ordinary skillin the art. For example, other devices, and forms of modularity, can beused.

Also, the inventors intend that only those claims which use the words“means for” are intended to be interpreted under 35 USC 112, sixthparagraph. Moreover, no limitations from the specification are intendedto be read into any claims, unless those limitations are expresslyincluded in the claims. The computers described herein may be any kindof computer, either general purpose, or some specific purpose computersuch as a workstation. The computer may be a Pentium class computer,running Windows XP or Linux, or may be a Macintosh computer. Thecomputer may also be a handheld computer, such as a PDA, cellphone, orlaptop.

The programs may be written in C, or Java, Brew or any other programminglanguage. The programs may be resident on a storage medium, e.g.,magnetic or optical, e.g. the computer hard drive, a removable disk ormedia such as a memory stick or SD media, or other removable medium. Theprograms may also be run over a network, for example, with a server orother machine sending signals to the local machine, which allows thelocal machine to carry out the operations described herein.

What is claimed is:
 1. An atomic force microscope, comprising: acantilever having a tip at one end and a driving part at its other end;a cantilever position detector; a positioning system for controlling aposition of the cantilever relative to the sample, the positioningsystem operating by: first operating by turning off a drive mechanismand positioning the cantilever at a specified point relative to thesample, and determining a stiffness of the cantilever from a measuredthermal spectrum taken far from a surface of the sample and recordinginformation indicative of the stiffness as representing a naturalfrequency of the cantilever; second approaching the tip of thecantilever to closer proximity with the sample and turning the drivemechanism on and setting a drive frequency of the cantilever to thenatural frequency of the cantilever as measured during said firstoperating, and approaching the tip of the cantilever to the surface ofthe sample until contact is established by setting the feedback setpoint to a desired interaction amplitude that is less than a freeoscillation amplitude; third operating by fully separating the samplefrom the tip of the cantilever and tuning the cantilever using the drivefrequency using a first relationship acquired; and after said tuning,turning off the drive mechanism and acquiring an additional thermalspectrum of the cantilever that is obtained during conditions that arecloser to imaging conditions; and repeating said first operating, andsecond operating using the additional thermal spectrum.
 2. Themicroscope as in claim 1, wherein said repeating comprises saidpositioning system using a simple harmonic oscillator model and datafrom the additional thermal spectrum to define the new natural frequencyof the cantilever and determine the phase-to-drive-frequencyrelationship of the cantilever; turning on the drive mechanism andchoosing a drive frequency based on measured cantilever parameters;choosing a drive amplitude which achieves the desired cantilever freeoscillation amplitude; and setting a driver phase from the result of theaforesaid simple harmonic oscillator model.
 3. The microscope as inclaim 2, wherein said measured cantilever parameters are equal either tothe new natural frequency of the cantilever or to a local amplitudemaximum of the cantilever near the cantilever resonant frequency.
 4. Themicroscope as in claim 1, wherein said Cantilever position detectorincludes a photodetector.
 5. The microscope as in claim 1, wherein saidpositioning system include a coarse and fine positioning system and acontroller, where said approaching and said inspection is determined byoptical inspection of the tip and the sample using thecoarse-positioning system.
 6. The microscope as in claim 1, wherein theposition detector includes a laser and a photodetector.
 7. Themicroscope as in claim 1, wherein said tuning the cantilever comprisesby either setting the drive frequency to the natural frequency of thecantilever as determined from the aforesaid thermal spectrum or settingthe drive frequency to a local amplitude maximum of the cantilever nearthe cantilever resonant frequency.